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Re: [xmca] a minus times a plus

Well I think it's a risky approach because I think it's
fortuitous that you get the right answer. There are only two
possible answers you had a 50-50 chance anyway!

In (-5) the - is firmly adhered to the 5.
I -[(5)*-7] the - applied to the result of the first

Applying the minus to one of the multiplicands is happens to
give the same answer as applying the minus to the product.
The laws of association for sense, i.e. A*(B*C) = (A*B)*C is
a later lesson surely!

Mike Cole wrote:
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 <ablunden@mira.net <mailto: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.


    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.,
        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
        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

        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
        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
        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;
        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
        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
        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?


From: Mike Cole [mailto:lchcmike@gmail.com
        <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 <mailto: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
        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
        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
        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
        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

        On Sun, May 3, 2009 at 9:32 AM, Tony Whitson <twhitson@udel.edu
        <mailto: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
        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
        basis in

        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
        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
        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: ematusov@udel.edu <mailto:ematusov@udel.edu>

        fax: 1-(302)-831-4110

        website:  <http://ematusov.soe.udel.edu/>

        publications:  <http://ematusov.soe.udel.edu/vita/publications.htm>

        Dialogic Pedagogy Forum:  <http://diaped.soe.udel.edu/>



        Tony Whitson
        UD School of Education
        NEWARK  DE  19716

        twhitson@udel.edu <mailto: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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