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*To*: mcole@weber.ucsd.edu, "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Subject*: Re: [xmca] a minus times a plus*From*: Jay Lemke <jaylemke@umich.edu>*Date*: Mon, 8 Jun 2009 00:27:08 +0200*Cc*:*Delivered-to*: xmca@weber.ucsd.edu*In-reply-to*: <30364f990906060912q78b96267of07e22bd33ad8283@mail.gmail.com>*List-archive*: <http://dss.ucsd.edu/mailman/private/xmca>*List-help*: <mailto:xmca-request@weber.ucsd.edu?subject=help>*List-id*: "eXtended Mind, Culture, Activity" <xmca.weber.ucsd.edu>*List-post*: <mailto:xmca@weber.ucsd.edu>*List-subscribe*: <http://dss.ucsd.edu/mailman/listinfo/xmca>, <mailto:xmca-request@weber.ucsd.edu?subject=subscribe>*List-unsubscribe*: <http://dss.ucsd.edu/mailman/listinfo/xmca>, <mailto:xmca-request@weber.ucsd.edu?subject=unsubscribe>*References*: <000c01c9cc09$551bb160$ff531420$@edu> <005201c9cc4e$0abf89d0$203e9d70$@edu> <49FE597C.60904@mira.net> <30364f990905251757q6cc33970q7fc3e68d4f18fee9@mail.gmail.com> <4A1B4675.9000906@mira.net> <30364f990905252018t30880eb0i49bbadfa5a04fc87@mail.gmail.com> <14a6419f0906030610l24f0909bv87242d42986fb875@mail.gmail.com> <3B19033D3E2EC34C97DF364119A79A61D33DF3@EXVS1.its.uidaho.edu> <30364f990906040858q55f40166ja4d9f339d0245618@mail.gmail.com> <14a6419f0906060311u590630f6oa50d5bd2e5f0c17d@mail.gmail.com> <30364f990906060912q78b96267of07e22bd33ad8283@mail.gmail.com>*Reply-to*: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Sender*: xmca-bounces@weber.ucsd.edu

JAY. Jay Lemke Professor Educational Studies University of Michigan Ann Arbor, MI 48109 www.umich.edu/~jaylemke On Jun 6, 2009, at 6:12 PM, Mike Cole wrote:

Hi Foo Keong-- It is so generous of you to even try to explain! Andyourquestion re math seems to me relevant to other areas of knowledge as well when you ask, "Can we condensefour thousand years ofhuman development into an easily digestible four minutes forlearners."Could we consider four years, just for whole numbers? Davydov startswithAlgebra as the gateway arithmetic. Jean Schmittau, Peter Moxhay andothersbelieve his method of introducing youngesters to math has some extrapower.As I understand it, others on xmca are dubious and look to othersources ofdifficulty. Karen Fuson, in her article on "developing mathematicalpowerins whole number operations" focuses on introducing number operations through very simple, familiar, imaginable, events where exchange is involved. Its odd to me experiencing the cycle of time, the "coming back to the beginning and recognizing itfor the first time" that is happening for me right now witharithmetic andearly algebra. The source is quite practical with social significance: the unbridgable gap the children I work with face betweenwhat their teachers are teaching about (say) subtraction (2005-118is mycurrent keystone example)trying to get their kids to learn that the first step is to subtract8 from15 and know enough to treat thesecond zero as a 9. But the child, even understanding that the tasktheteacher is focused on is disabled because when asked 15-8 the answer =3 and only painstakingattention to the problem set up with fingers and subtracting one byone,with full compliance and even eagerness by the child, brings her to 7.Now suppose this phenomenon is ubiquitous, affects 100's ofthousands ofchildren, and is heavily correlated with social class. Then .... ??? ....I think my frustration is probably equivalent to yourse inintensity, butthe quality is of a somewhat different nature. mikeOn Sat, Jun 6, 2009 at 3:11 AM, Ng Foo Keong <lefouque@gmail.com>wrote:I was trained in mathematics at the University of Cambridge (UK) for my undergraduate studies, concentrating more on pure mathematics (including algebra). I am able to roll out a rigorous abstract proof of why "minus times minus" is a "plus", using only the basic axioms of real numbers (actually you only need a few of those axioms). However, abstract proofs aren't likely to be useful for non-math specialists and struggling neophyte learners of algebra. in order to pull off such a proof, or even just to understand just the few lines of proof, you almost need to be a mental masochist. Who likes to go through mental torture? Can we condense four thousand years of human development of mathematical understanding into an easily digestible four minutes for learners? thus the huge gulf of understanding still persists. that's why as an educator, i feel so useless being unable to help other people. :-( F.K. 2009/6/4 Mike Cole <lchcmike@gmail.com>:I am currently reading article by Fuson suggestion by Anna Sfardon wholenumber operations. I also need to study Anna's paper with exactlythisexample in it. Not sure what moment of despair at deeperunderstandinghitme. Now that I am done teaching and have a whole day to communicatethingsare looking up!! Apologies for doubting I could have deepunderstandingofwhy minus x minus = plus and minus x plus = minus. At present myunderstanding remains somewhat bifurcated. The former is negationof anegation as david kel long ago suggested, linking his suggestion toAnna'scomognition approach. The second I think more of in terms of number line and multiplication as repeated addition. Perhaps the two will coalesce under your combined tutelage. mike And member book links are coming in. Nice. mike_______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca

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**Follow-Ups**:**Re: [xmca] a minus times a plus***From:*Mike Cole <lchcmike@gmail.com>

**References**:**Re: [xmca] a minus times a plus***From:*Ng Foo Keong <lefouque@gmail.com>

**RE: [xmca] a minus times a plus***From:*"Duvall, Emily" <emily@uidaho.edu>

**Re: [xmca] a minus times a plus***From:*Mike Cole <lchcmike@gmail.com>

**Re: [xmca] a minus times a plus***From:*Ng Foo Keong <lefouque@gmail.com>

**Re: [xmca] a minus times a plus***From:*Mike Cole <lchcmike@gmail.com>

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