Re: [xmca] a minus times a plus

[Mike Cole <lchcmike@gmail.com>]

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.

mike

2009/5/3 Andy Blunden <ablunden@mira.net>

> 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.
>
> Andy
>
> Eugene Matusov wrote:
>
>> 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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> --
> ------------------------------------------------------------------------
> Andy Blunden http://home.mira.net/~andy/ <http://home.mira.net/%7Eandy/>
> Hegel's Logic with a Foreword by Andy Blunden:
> From Erythro's Press and Media <http://www.erythrospress.com/>.
>
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