# Re: [xmca] a minus times a plus

```I guess i should give it up and go fishing. I am a terrible mathematician
and should probably just give it, and the kids, up. Sad, but probably true.
ike

On Mon, May 25, 2009 at 6:31 PM, Andy Blunden <ablunden@mira.net> wrote:

> Well I think it's a risky approach because I think it's
> fortuitous that you get the right answer. There are only two
>
> In (-5) the - is firmly adhered to the 5.
> I -[(5)*-7] the - applied to the result of the first
> multiplication.
>
> 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!
>
> Andy
> 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
>> 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 <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
>>    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
>>        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
>>        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,
>>        (-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
>>        (-1)*(+1)=-1.
>>        Again, it is not always true. For example, sometimes when we are
>>        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
>>        <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
>>        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
>>        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
>>        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
>>        <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
>>        (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 <mailto: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 <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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>>
>>    --
>>  ------------------------------------------------------------------------
>>    Andy Blunden http://home.mira.net/~andy/<http://home.mira.net/%7Eandy/><
>> 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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