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*To*: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Subject*: Re: [xmca] a minus times a plus*From*: Steve Gabosch <stevegabosch@me.com>*Date*: Sun, 28 Jun 2009 12:02:12 -0700*Delivered-to*: xmca@weber.ucsd.edu*In-reply-to*: <731CECC23FB8CA4E9127BD399744D1EC01A3409F@email001.lsu.edu>*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> <4A470700.40206@mira.net> <D6626823-48CA-4CC1-B580-B5C8346CA5A5@me.com> <731CECC23FB8CA4E9127BD399744D1EC01A3409F@email001.lsu.edu>*Reply-to*: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Sender*: xmca-bounces@weber.ucsd.edu

You might be right, David. Here are some thoughts ...

First, the idea of internalizing auxiliary stimuli.

Cheers, - Steve On Jun 28, 2009, at 7:36 AM, David H Kirshner wrote:

Steve,Whereas the advice seems sound, it strikes me as much more in thePiagetian/constructivist genre than in the Vygotskyan/socioculturalgenre. Do we make any educational use of Vygotsky's insight thatlearning comes about through internalization of social processes, oris this just a general characterization of how developmentprogresses. When it comes time to teach specific conceptual content(rather than broad methods) we become Piagetians.David -----Original Message-----From: xmca-bounces@weber.ucsd.edu [mailto:xmca-bounces@weber.ucsd.edu] On Behalf Of Steve GaboschSent: Sunday, June 28, 2009 8:58 AM To: eXtended Mind, Culture, Activity Subject: Re: [xmca] a minus times a plus Andy, the following from Leontiev I happened to be reading tonight spurs a thought that might pertain to kids and adults when they get "stuck" when learning. Maybe this has some relevance to the situation you are speaking of. Just stabbing in the dark, of course. Interesting idea in any case. Leontiev talks about returning students to an **earlier** stage of a learning process when they may have not have yet fully reorganized their use of external operations into the new kinds of internal mental required for the next stage. One step backwards creates the possibility for steps forward, so to speak. These paragraphs are from your new version of Leontiev's book, Development of Mind, pages 392-393, at the end of the last essay, The Child's Psychological Development and Mental Deficiency. As the notes explain, this is the text of a lecture AN Leontiev gave to a World Health Organization seminar in Milan in 1959. I find this to be an interesting idea, that sometimes we need to go backwards and reorganize old methods before being able to competently proceed to new processes. Hmmm. Come to think of it, this sounds like something I wind up doing a lot! LOL Leontiev: "To explain what I mean, let me cite a simple experiment I once made in a school for mentally backward children. "I drew attention to the fact that the pupils, while doing mental addition, were using their fingers for it in a concealed way. Then I asked for several saucers, gave two to each pupil, and told them to hold them above the desk while they were giving their answers. In these conditions it proved that the operation of adding numbers broke down completely in most of them. More detailed analysis indi-cated that these children had in fact remained at the stage, asregardsaddition, of the external operations of 'counting by ones', and had not passed to the next stage. They therefore could not advance in learning arithmetic beyond actions within the first ten numbers with- out special help. For that purpose it was necessary, not to take themfurther, but on the contrary to return them first to the originalstageof developed external operations, to 'reduce' these operations prop- erly and to transfer them to the oral plane, in short to build a capacity 'to count in their head' all over again. "Research has shown that such a reorganisation is actually possible even when working with children of quite pronounced mental back- wardness. It is specially important that this approach has the effect, in cases of a slight lag in mental development, of completely eliminating it. "Such intervention in the forming of mental operations of some kind or other must, of course, be prompt and timely, because other- wise the forming of the process cannot proceed further normally be- cause the stage of its forming has sometimes not been built up by chance or has been built up incorrectly, with the result that an im- pression of alleged mental incapacity in the child is created." <end of quote> - Steve On Jun 27, 2009, at 11:00 PM, Andy Blunden wrote:I hope people won't mind if I continue to pick the brains of this list on the problem of my niece's progress in maths, or lack of it. It seems that the suggestion last time - that Marissa may have missed important lessons while on holiday - may explain her poor performance last year in maths, even though maths has always been her weak subject. She has caught up a bit but she is still badly behind. It seems that the issue Mike has raised also applies: she is getting homework that seem to presume she know things that in fact she doesn't. The only other negative in her school reports is that she doesn't participate in class discussion or ask questions when she doesn't understand something. I presume the hesitancy about speaking up is probably the cause of failure to correct her maths problems and the teachers giving her homework she doesn't understand. She is now 15 and her maths homework is also beyond her father! :) and the crisis of the transition from childhood to adulthood around this age, makes it impossible for the father to get Marissa talk about it to him, or engage Marissa in games of 20 Questions or something to lead her to the joys of asking others. Discussion over the dinner table is apparently also unconducive to her participation. Does anyone have any ideas? I've run out of suggestions. I could probably help if I was there, but I'm 1000 km away. 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. mikeOn 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 sourceis quite practical with social significance: the unbridgable gapthechildren 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-- ------------------------------------------------------------------------ Andy Blunden (Erythrós Press and Media) http://www.erythrospress.com/ Orders: http://www.erythrospress.com/store/main.html#books _______________________________________________ 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 _______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca

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**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>

**Re: [xmca] a minus times a plus***From:*Andy Blunden <ablunden@mira.net>

**Re: [xmca] a minus times a plus***From:*Steve Gabosch <stevegabosch@me.com>

**RE: [xmca] a minus times a plus***From:*"David H Kirshner" <dkirsh@lsu.edu>

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