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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
> 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
> 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
>> 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 <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.
>>
>> 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
>> <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
>> 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
>> <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)
>>
>> __________ Information from ESET NOD32 Antivirus,
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>> virus signature
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>>
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>>
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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:
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>>
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