Andy-- Until getting back to this exchange, i had made it only as far as:
5* (-7) is (-7 seven times down the line) which seems pretty easy to
represent and communicate about.
But when we move to (-5) * (-5) I can understand it better myself (hah!)
if I rewrite the problem as
-[(5)*-7]. So inside the bracket I do what i did in the 5*(-7) example
and then treat - as "do the opposite," an operator.
I am sure this is all lousy thinking, but that is how far I have gotten.
2009/5/3 Andy Blunden <firstname.lastname@example.org <mailto:email@example.com>>
Thanks for those observations Eugene. Can I just throw a couple of
things on to the heap?
Your translation of разпредмечивание as de-objectivization is right
I think. Someone I have read recently, maybe Kozulin, made a big
deal of Vygotsky's use of the idea, and also ascribed it to Hegel,
which I am not at all sure about.
As a civil engineering student we were taught to imagine ourselves
as a building. Complex structures are formally indeterminate
mathematically, you have to use successive approximation to
calculate stresses and even then the point is to design in advance,
not calculate afterwards. So far and away the best approach is to
imagine yourself as the building and "feel" where the stresses are
and how you have to adjust your position to bear the weight, and
then sketh it in in steel.
But how to de-objectivize -x-=+? I am actually of the view that it
is better to postpone the justification of the rule till after a
child has had experience in using it, but I am not the teacher here.
So I wouldn't try explaining the deeper meaning of multiplication
which unites it with various non-arithmetic operations until after
the child can multiply arithmetically. Arithmetic is actually the
richest domain in all of mathematics; all other domains are
sub-parts of arithmetic! Learn arithmetic and everything else opens
to you. But ....
Multiplication is compounding. It is "of" rather than "and". In
primary school we were actually taught "of" as an additional
operations over and above "multiply". Odd.
So -2x-3 is -2 of -3, whereas -2+-3 is -2 and -3. So if a child is
linguistically well-developed, that might help.
Eugene Matusov wrote:
Let me try to tackle yours and Sophie's math problem since I'm very
sympathetic to it, "I am working and thinking about Sophie's
to understand -2*6. The use of multiplication as repeated
but when I get to -2*-6 I feel as if I am only part way there
something like Jerry's mirror approach."
1) In my view, to understand a math model and a math
problem means to
subjectivize it - namely to translate it back to the bodily
social relations. There is a useful Russian term
"разпредмечивание" that I
do not know how to translate (de-objectivization?). A person has
to find a
human experience ("переживание"), in which the math model and
make sense for the person. Nunes talks about "embodied
cognition" - I like
this term. Dividing pizza on equal parts is an example of such
subjectivization of fractional division. When I was in high
realized that calculus is "geometry for blind people" - it
really helped me
to understand bizarreness of calculus. The problem is to find such
subjectivizition for -2*-6.
2) In math, the minus represents undesired human values
debt, enemies, hole, absence, past, death, decay, giving away,
prison, and so on, while plus represents desired human values
income, friends, surplus, presence, future, life, growth,
rich, freedom and so on. Of course, these values can be relative
person: what is good for one is bad for another and vice versa.
also relative to cultures:
3) In math, the procedure of multiplication usually means
"application". For example, 2 multiply by 3 means that each of
the 2 Units
(e.g., people, places, boxes) we apply (=give) 3 unites (e.g.,
objects, dollars). The questions is how many unites we have now.
4) Application of "good" to "good" (i.e., multiplication of
numbers) is always good in the math model (+1)*(+1)=+1, which is
true in the reality. For example, kind people are good, eating
is good as
well, however, if we apply too much eating to kind people, the
result is not
necessary good because too much eating might lead to obesity,
which is bad
(-1), thus, (+1)*(+1)=-1. Mathematical model ALWAYS have limited
we should watch out for how we use them. However, there are
might fit our mathematical models and thus mathematical models
5) If we apply good to the bad, the bad will increase
Again, it is not always true. For example, sometimes when we are
kind to bad
people, they soften and become kinder, not worse, thus,
(-1)*(+1)=+1. But in
many cases, they become worse as the math model predicts. For
Western nations were kinder to Hitler's Germany, it became more
dangerous (worse). If you help (+1) to bad side (-1), it is
in making bad things (=-1).
6) Similarly, if you apply bad to the good, the good
(+1)*(-1)=-1. As you expect, it is not always true. Taking dramatic
examples, when some good people are wrongly accused and get to
jail, some of
them became stronger spiritually (e.g., boxer Hurricane) - in
(+1)*(-1)=+1. But in many cases, when bad things are applied to
the good usually suffers (-1), what the math model predicts.
7) Finally, when bad is applied to the bad (-1)*(-1), it
weakens the bad and strengthens the good (-1)*(-1)=+1. For
(-1) of your enemy (-1) can become your ally (+1). Or in
(-1) is applied to death (-1) creates the life of resurrection (+1).
Punishment (-1) of a criminal (-1) is retribution=justice (+1).
mathematical model does not always work: enemy of your enemy can
your enemy; death applied to death might result in a zombie;
applied to a criminal might lead to hardening his or her heart
recidivism (in all these example, (-1)*(-1)=-1). ALL
have limitations and we should be careful in using them and
they might stop working for us and our objects. Even as familiar
as 2+2=4 do not work always: two friends plus two friends are
four friends! (for my family, 1+1=3, my wife and I have one son ;-).
8) So, here are several of my subjectivizations of -2*-6:
a. Each of your two enemies (-2 for you) has six their own
for your enemies). How many potential allies you might have?
b. Sad reality but for long time, Eugene has been paying $2
to a bank a
year (-2 for Eugene) for his college debt (alas!). How richer
was Eugene six
years ago (-6 years)? Negative income (=debt) times negative
equals past treasure:. (This is a heartbreaking math task for me!)
c. On more optimistic note, when I put my yogurt into my
temperature drops 2 degrees each hour (-2 degrees for yogurt).
How warmer my
yogurt was 6 hours ago (-6 hours)?
9) Thinking about a minus times a minus multiplication, I
found that it
is less common for our everyday experiences than many other math
I have developed many examples but they were so contrived that
wonder it is not math for life but life for math:
Mike, I wonder if you organize your discussion with Sophie
subjectivizations and limitations of math models, it might help
her. Let me
know if you decide to do that: I wonder if there are other and
subjectivizations of (-1)*(-1)=1:. Of course, there is a pure
that -2*-6=12 but I'm not sure it can be useful for Sophie.
What do you think?
From: Mike Cole [mailto:firstname.lastname@example.org
<mailto:email@example.com>] Sent: Sunday, May 03, 2009 2:38 PM
To: Tony Whitson
Cc: Eugene Matusov; eXtended Mind, Culture, Activity; PIG;
Subject: Re: [UD-PIG] What good for kids seems dangerous for adults
Eugene, Tony, et al.
Firstly, I would like to follow up with the discussion of
binaries which I
think is important, and allied items that came up in those
notes. But Eugene
I can do that off line or when we (finally!) get to see each
whenever. Unless the issues are of import to others who would seek
tell us how we are both wrong headed, or whatever. I also want
seriously about the issue of youth desired activities and adult
activities as these influence our work and general
understanding. But this
is also a large issue and will take time and should not be
of narrow interest. So I would prefer to hear other voices chime
in, as has
happened incredibly with the minus/plus math discussion.
(Another version of "what do you all think" rented from Eugene).
And a way
of dealing with urgent need to respond to a very large number of
fieldnotes before morning!!!).
Tony-- Your take on the issue Eugene raised is not what we are
about, but not unrelated. To me a really major manifestation of the
you are writing about is that in 1983 Sheila and I could write a
Barker and Wright's *One Boy's Day* was relevant, if antique.
But you will
not find that empirical example (nor a lot else) in the current
that textbook. I rode the streets of LA and climbed around its
at a kid, and sold papers on a street corner in west LA in the
when "Midwest" was still a going mid-western town. NO NO NO now.
fashioned it might make the current generation titter as they
on that later.
I am working and thinking about Sophie's brave efforts to
The use of multiplication as repeated addition helps, but when I get
to -2*-6 I feel as if I am only part way there and want
Jerry's mirror approach. What makes it so strange is that at
I have no trouble with the contents of figure 1. Something about
commognition going it seems. Gotta study Ng's pic too.
Now, gotta go back to my local students until I have given them
they need for this coming week of work/learning/fun. Kotbegmot
will be back here with you-all ere too long
On Sun, May 3, 2009 at 9:32 AM, Tony Whitson <firstname.lastname@example.org
I am eager to hear what Mike has to say.
At the risk of commenting on something that may be different
and/or Eugene's meaning:
I think this has become more and more prevalent over the course
lifetime, at least in the US.
I went to school through 12th grade in Iowa, where there wasn't
go, really, even after age 16 when you could drive (although
there were all
kinds of adventures possible by bicycle).
When I moved to Boston at 18, one thing that seemed really
exciting to me
was the way kids had free reign of that marvellous city,
the MTA. When I lived in Chinatown, I saw diverse groups of kids
for example, Chinese and Italian from the North End) freely
roaming the city
on the Boston subway system.
That seemed to change at the time of the conflict over busing, when
politicians like Louise Day Hicks
fanned the flames of fear and suspicion among population groups.
Then, of course, came the paranoia over "Mr. Stranger Danger"
-- which although perhaps overreaction, was not totally without
Now I live in an apartment complex with one entry from a
suburban street to
the lanes and parking lots within our complex. School buses pick
kids up and
drop kids off at that entry. At an age when I was riding my bike
town in Illinois and then in Iowa, the kids today are watched
over by their
parents until they're on the bus, and then greeted by parents
them when they're dropped off when they get home.
I expect that Eugene and probably Mike were referring to things
meaningful intellectually, aesthetically, etc.; but I think the
the US at least, goes way beyond that.
What do you think?
On Sun, 3 May 2009, Eugene Matusov wrote:
Many years ago, you made a very good point in one of our private
conversations that unfortunately, I did not write down after
you. You said
something like, "Often what is meaningful for kids seems to be
adults." Is my memory correct? Can you elaborate on that? Have
By now, I have so many observations and examples of this sad
point. I wish
somebody studied this phenomenon on a systematic basis. I saw so
when adults literally suck the life out of kids because of their
about kids' safety and well-being.
What do you think?
Eugene Matusov, Ph.D.
Professor of Education
School of Education
University of Delaware
Newark, DE 19716, USA
email: email@example.com <mailto:firstname.lastname@example.org>
Dialogic Pedagogy Forum: <http://diaped.soe.udel.edu/>
UD School of Education
NEWARK DE 19716
"those who fail to reread
are obliged to read the same story everywhere"
-- Roland Barthes, S/Z (1970)
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