Dear Mike-
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 brave efforts
to understand -2*6. 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
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 experiences and
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 the problem
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 school, I
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 (bad) like
debt, enemies, hole, absence, past, death, decay, giving away, cold, poor,
prison, and so on, while plus represents desired human values (good) like
income, friends, surplus, presence, future, life, growth, receiving, hot,
rich, freedom and so on. Of course, these values can be relative to a
person: what is good for one is bad for another and vice versa. They are
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., applies,
objects, dollars). The questions is how many unites we have now.
4) Application of "good" to "good" (i.e., multiplication of positive
numbers) is always good in the math model (+1)*(+1)=+1, which is not always
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 power and
we should watch out for how we use them. However, there are objects that
might fit our mathematical models and thus mathematical models can be
helpful.
5) If we apply good to the bad, the bad will increase (-1)*(+1)=-1.
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 example, while
Western nations were kinder to Hitler's Germany, it became more powerful and
dangerous (worse). If you help (+1) to bad side (-1), it is getting stronger
in making bad things (=-1).
6) Similarly, if you apply bad to the good, the good becomes worse
(+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 these cases,
(+1)*(-1)=+1. But in many cases, when bad things are applied to the good,
the good usually suffers (-1), what the math model predicts.
7) Finally, when bad is applied to the bad (-1)*(-1), it usually
weakens the bad and strengthens the good (-1)*(-1)=+1. For example, enemy
(-1) of your enemy (-1) can become your ally (+1). Or in Christianity, death
(-1) is applied to death (-1) creates the life of resurrection (+1).
Punishment (-1) of a criminal (-1) is retribution=justice (+1). Again this
mathematical model does not always work: enemy of your enemy can still be
your enemy; death applied to death might result in a zombie; punishment
applied to a criminal might lead to hardening his or her heart and to
recidivism (in all these example, (-1)*(-1)=-1). ALL mathematical models
have limitations and we should be careful in using them and explore when
they might stop working for us and our objects. Even as familiar math model
as 2+2=4 do not work always: two friends plus two friends are not always
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 enemies (-6
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 time (=past)
equals past treasure:. (This is a heartbreaking math task for me!)
c. On more optimistic note, when I put my yogurt into my freezer, its
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 procedures.
I have developed many examples but they were so contrived that one would
wonder it is not math for life but life for math:
Mike, I wonder if you organize your discussion with Sophie around these
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 better
subjectivizations of (-1)*(-1)=1:. Of course, there is a pure math proof
that -2*-6=12 but I'm not sure it can be useful for Sophie.
What do you think?
Eugene
From: Mike Cole [mailto:lchcmike@gmail.com]
Sent: Sunday, May 03, 2009 2:38 PM
To: Tony Whitson
Cc: Eugene Matusov; eXtended Mind, Culture, Activity; PIG;
backontrack@wwscholars.org; Zoi Philippakos
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
and
I can do that off line or when we (finally!) get to see each other, or
whenever. Unless the issues are of import to others who would seek
clarification or
tell us how we are both wrong headed, or whatever. I also want to write
seriously about the issue of youth desired activities and adult sanctioned
activities as these influence our work and general understanding. But this
is also a large issue and will take time and should not be discussed if
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 student
fieldnotes before morning!!!).
Tony-- Your take on the issue Eugene raised is not what we are talking
about, but not unrelated. To me a really major manifestation of the
phenomenon
you are writing about is that in 1983 Sheila and I could write a text where
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 version of
that textbook. I rode the streets of LA and climbed around its sewer system
at a kid, and sold papers on a street corner in west LA in the late 40's
when "Midwest" was still a going mid-western town. NO NO NO now. So old
fashioned it might make the current generation titter as they twitter. More
on that later.
I am working and thinking about Sophie's brave efforts to understand -2*6.
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 something like
Jerry's mirror approach. What makes it so strange is that at another level
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 the feedback
they need for this coming week of work/learning/fun. Kotbegmot willing, I
will be back here with you-all ere too long
mike
On Sun, May 3, 2009 at 9:32 AM, Tony Whitson <twhitson@udel.edu> wrote:
I am eager to hear what Mike has to say.
At the risk of commenting on something that may be different from Mike
and/or Eugene's meaning:
I think this has become more and more prevalent over the course of my
lifetime, at least in the US.
I went to school through 12th grade in Iowa, where there wasn't anyplace to
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, inexpensively via
the MTA. When I lived in Chinatown, I saw diverse groups of kids (mixing,
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
http://en.wikipedia.org/wiki/Louise_Day_Hicks
fanned the flames of fear and suspicion among population groups.
Then, of course, came the paranoia over "Mr. Stranger Danger"
http://en.wikipedia.org/wiki/Stranger_danger
-- which although perhaps overreaction, was not totally without basis in
reality.
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 all over
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 waiting for
them when they're dropped off when they get home.
I expect that Eugene and probably Mike were referring to things that are
meaningful intellectually, aesthetically, etc.; but I think the problem, in
the US at least, goes way beyond that.
What do you think?
On Sun, 3 May 2009, Eugene Matusov wrote:
Dear Mike-
Many years ago, you made a very good point in one of our private phone
conversations that unfortunately, I did not write down after you. You said
something like, "Often what is meaningful for kids seems to be dangerous for
adults." Is my memory correct? Can you elaborate on that? Have ever written
on that?
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 many cases
when adults literally suck the life out of kids because of their concerns
about kids' safety and well-being.
What do you think?
Eugene
---------------------
Eugene Matusov, Ph.D.
Professor of Education
School of Education
University of Delaware
Newark, DE 19716, USA
email: ematusov@udel.edu
fax: 1-(302)-831-4110
website: <http://ematusov.soe.udel.edu/> http://ematusov.soe.udel.edu
publications: <http://ematusov.soe.udel.edu/vita/publications.htm>
http://ematusov.soe.udel.edu/vita/publications.htm
Dialogic Pedagogy Forum: <http://diaped.soe.udel.edu/>
http://diaped.soe.udel.edu
---------------------
Tony Whitson
UD School of Education
NEWARK DE 19716
twhitson@udel.edu
_______________________________
"those who fail to reread
are obliged to read the same story everywhere"
-- Roland Barthes, S/Z (1970)
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