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Re: [xmca] a minus times a plus

You might be right, David.  Here are some thoughts ...

The Vygotskyan content of the idea touched on by Leontiev occurs to me along two lines. I certainly don't know if either of these are at all relevant to the situations Andy or Mike have been speaking of. Just throwing out ideas.

First, the idea of internalizing auxiliary stimuli.

Correct me if I am wrong, but I am thinking that Leontiev's idea as quoted corresponds to ideas that Vygotsky describes in Mind and Society. For example, on pg 45, he says "Our results ... indicate three basic stages in the development of mediated remembering. At the first stage (preschool age) the child is not capable of mastering his behavior by organizing special stimuli .... [In] The second stage of development ... [for example, as the experiments revealed] The introduction of cards as a system of auxiliary, external stimuli raises the effectiveness of the child's activity considerably. ... At this stage the external sign predominates ... At the third stage (among adults) ... the auxiliary stimuli are emancipated from primary external forms."

My question to you, David, while this idea seems to be Vygotskyan, is it also Piagetian?

Second, the idea of rising from concrete complexes (everyday concepts) to abstract, academic concepts.

The other day I was re-reading Thinking and Speech, chapter 6 section 6, taking another look at Vygotsky's metaphor about the globe overlaid with a matrix of equivalent concepts (going east to west) and increasingly abstract or concrete concepts (going north to south). This may be a nice way to relate Leontiev's point. Let's see if it works.

Vygotsky uses this globe metaphor or model to explain the law of equivalent concepts "**any concept can be represented through other concepts in an infinite number of ways**.” pg 226 Minick. (This I believe is the "law" behind the point Roger just made in his message, when he said "I'd look to see how your daughter has learned in the areas in which she has been successful and look for possible analogous methods that might apply to mathematics." That is utilizing the law of equivalence Vygotsky spends some time describing with this globe model.

This is part of the **relations of generality** Vygotsky emphasizes, that is, the underlying capacity for all concepts to be related to one another along two kinds of dimensions - 1) relative abstractness and concreteness, and therefore, 2) equivalance of generality. This creates a third kind of underlying relationship as a child passes from syncretic formations to complexes, to preconcepts to true concepts - 3) new kinds of mental operations, at each stage of now reorganization of that individual's concept formation system.

In this global model, I think Vygotsky puts the most extremely graphic, concrete concepts at the South pole (it takes a little doing to get this model straightened out - as discussed here on xmca in the past, Vygotsky has the terms for longitude and latitude reversed, and there is a little confusion over whether he is putting the concrete at the South or the North, too).

Using this model, if in some area, such a maths, a student is "stuck" down "South" in concepts that are concrete, graphic, and in perhaps external-stimuli oriented, they may be having trouble moving "up" North to the more abstract levels of generalization required to master the academic material.

Vygotsky uses the example of thinking concretely in graphic "ones" (this would be South pole type thinking) versus thinking abstractly in terms of the infinite number of ways the concept "one" can be conceptualized arithmetically, such as subtracting 999,999 from 1,000,000 or taking the difference of any two adjacent numbers. On the globe, this would be moving Northward toward a much higher level of abstraction about numbers).

Now, revisiting Leontiev's story about the students who held the saucers but couldn't count without using their fingers ... until the students revisited the counting process and transferred what they could do with their fingers to the oral plane.

If we look at the globe, perhaps the "equivalent concept" (perhaps more precisely, equivalent complex) of counting by ones on fingers and counting by ones orally was the necessary transitional transformation. In Vygotsky's model, the student's were taught how to shift first in an east-west direction (latitudinally) by going from fingers to spoken words. And then eventually they could learn to move up and down the new north-south longitudinal line that now employs new kinds of tools, signs, concepts - spoken word-numbers.

Here is an illustration, which may or may not be valid. This is an exercise that might work with a child of that stage of learning might look like this. Has anyone ever done this? Something in the back of my mind tells me I may have learned numbers using something like this. First, of course, are the fingers. And then there is memorizing numbers and using fingers and numbers together. So far so good, this where the students Leontiev speaks of were at. But here is the transition. Add a new step to just counting and adding fingers. Say "2 plus 1 equals 3." Then say "3 plus 1 equals 4." And on an on. Stick to adjacent numbers and make up games, rhymes, etc. using this pattern. It can be quite rhythmic. And it is a remarkable rule about arithmetic to learn. It never fails! After a while, the fingers aren't needed. You can do this holding saucers. Once the rule about "plus one" is mastered, try plus two. "2 plus 2 equals 4." "3 plus 2 equals 5." "4 plus 2 equals 6." It works every time! And of course, there is minus ...

The idea here is that a new kind of external stimulus is substituted for the old one. In this case, the new stimulus requires a new kind of mental operation (speech operations) that can replace the finger stimuli. This creates a new north-south line to move on, which the fingers operations would not permit.

And so we go, hopping around the globe as best we can, starting from places we have begun to master and traveling to new territories via equivalent concepts, varying unities of abstractness and concreteness, and the necessary mental operations to go along with the individual's stage of conceptual development. This is Vygotsky's description. He called this "global" process the "measure of generality".

When moving about this globe of possible human concepts, we can learn to move not only to equivalent concepts, east-west, but also begin to master ever more abstract concepts, moving northward.

What makes Leontiev's idea so interesting is that sometimes, to move northward, we may need to go southward first! We may have to do that to find an east-west route that is available to us, and this may be the road to getting unstuck.

Does this work? Is this a helpful way for a teacher, parent, student helping another student, worker helping another worker, to grasp ways to convey to another new ways to internalize a new concept?

So am I thinking along Vygotskyan lines ... or am I stuck in a Piagetian paradigm ... seriously, maybe David is seeing something I am not ... or are we by nature Piagetians when we teach specific conceptual content? ...

- Steve

On Jun 28, 2009, at 7:36 AM, David H Kirshner wrote:

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.

-----Original Message-----
From: xmca-bounces@weber.ucsd.edu [mailto:xmca- bounces@weber.ucsd.edu] 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 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
'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>

- 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

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 <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
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
(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
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
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
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

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
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
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>

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

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
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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