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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*: Andy Blunden <ablunden@mira.net>*Date*: Sun, 28 Jun 2009 16:00:32 +1000*Cc*:*Delivered-to*: xmca@weber.ucsd.edu*In-reply-to*: <30364f990906071537q7a210931x814cd0995e6d06fe@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> <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> <819F9838-8576-4C9D-8464-6808D856DC58@umich.edu> <30364f990906071537q7a210931x814cd0995e6d06fe@mail.gmail.com>*Reply-to*: ablunden@mira.net, "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Sender*: xmca-bounces@weber.ucsd.edu*User-agent*: Thunderbird 2.0.0.14 (Windows/20080421)

Andy Mike Cole wrote:

SO glad you are interested in this, Jay. I have just made contact with Karen Fuson who has, lucky for us, "retired" from Northwestern and moved nearby. She is away for a week or so but then we are getting together. This is a problem that just may be tractable, theoretically interesting for sure, attractive of experience collaborators, and god knows, of practrical importance to lots of kids. mike On Sun, Jun 7, 2009 at 3:27 PM, Jay Lemke <jaylemke@umich.edu> wrote:Yes, Mike and F.K., these are very disturbing issues. Both that what we think we want to teach seems to depend on deeper (e.g. 4000-year deep) knowledge than it's realistic to expect most people to learn (or want to learn), and that how we teach even the most practical bits of mathematics (like 15 minus 8) seems to have gone so wrong that it's hard to know where to start, especially for those we have most systematically failed. We do indeed need to not give up. But we also need, I think, to admit that it's time to seriously re-think the whole of the what, why, and how of education. Math is a nice place to focus because at least some of it seems universally agreed to be useful by almost everyone, because professional mathematicians and most people, including teachers and mathematics educators, seem to hold radically different views about what the subject is, and because success in teaching it, measured in almost any way, is pretty near the bottom of the heap. Yes, we can find somewhat better ways to teach the same stuff, but maybe it's the stuff itself (the content of the curriculum, viewed not just as information, but as activity) that needs to be rethought? along with the ethics and efficacy of who decides. No matter how many times you multiply a minus by any number of pluses, you still get a minus. 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! And your question re math seems to me relevant to other areas of knowledge as well when you ask, "Can we condensefour thousand years of human development into an easily digestible four minutes for learners." Could we consider four years, just for whole numbers? Davydov starts with Algebra as the gateway arithmetic. Jean Schmittau, Peter Moxhay and others believe his method of introducing youngesters to math has some extra power. As I understand it, others on xmca are dubious and look to other sources of difficulty. Karen Fuson, in her article on "developing mathematical power ins 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 it for the first time" that is happening for me right now with arithmetic and early algebra. The source is quite practical with social significance: the unbridgable gap the children I work with face between what their teachers are teaching about (say) subtraction (2005-118 is my current keystone example) trying to get their kids to learn that the first step is to subtract 8 from 15 and know enough to treat the second zero as a 9. But the child, even understanding that the task the teacher is focused on is disabled because when asked 15-8 the answer =3 and only painstaking attention to the problem set up with fingers and subtracting one by one, with full compliance and even eagerness by the child, brings her to 7. Now suppose this phenomenon is ubiquitous, affects 100's of thousands of children, and is heavily correlated with social class. Then .... ??? .... I think my frustration is probably equivalent to yourse in intensity, but the quality is of a somewhat different nature. mike On 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 Sfard on whole number operations. I also need to study Anna's paper with exactly this example in it. Not sure what moment of despair at deeper understanding hit me. Now that I am done teaching and have a whole day to communicate things are looking up!! Apologies for doubting I could have deep understanding of why minus x minus = plus and minus x plus = minus. At present my understanding remains somewhat bifurcated. The former is negation of a negation as david kel long ago suggested, linking his suggestion to Anna's comognition 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_______________________________________________ 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:*Steve Gabosch <stevegabosch@me.com>

**Re: [xmca] a minus times a plus***From:*Shirley Franklin <s.franklin@dsl.pipex.com>

**Re: [xmca] a minus times a plus***From:*Ivan Rosero <irosero@ucsd.edu>

**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>

**Re: [xmca] a minus times a plus***From:*Jay Lemke <jaylemke@umich.edu>

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

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