[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]

*To*: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Subject*: Re: [xmca] a minus times a plus*From*: Mike Cole <lchcmike@gmail.com>*Date*: Fri, 24 Jul 2009 17:50:22 -0700*Delivered-to*: xmca@weber.ucsd.edu*Dkim-signature*: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=gamma; h=domainkey-signature:mime-version:received:reply-to:in-reply-to :references:date:message-id:subject:from:to:content-type; bh=Yv+CmZwc9QyHk+0+LibFPoR+kJB089knVxRCvzY2n84=; b=gXPNdMFFt13XNYEbnpGbXDmiYU9fg5UvJxZRbCXwGyRpiwyvI2bQWyxST5f83G0Zis 9V/4HdGcsVSd3S9CiRZgBtaqvY629Man5+dVSY6/qVlBSKia7ZqAwvJwBPWDXjVpGobb 3w4Co0fjWRjedMTRGh6QwKh0yk+XIhYWD/L+Y=*Domainkey-signature*: a=rsa-sha1; c=nofws; d=gmail.com; s=gamma; h=mime-version:reply-to:in-reply-to:references:date:message-id :subject:from:to:content-type; b=v6JqbfO224r9wkG/Z6EG48jAfR2t8N77n+QqXJJCv8ACA2VriMHc/cgF9Sx3SIFJNI WOWWh7PFEVxHmZmNXX14exZsH9ivzhNQnKFt5BknQZF3OPuy7xFM2/reETHKCG2pd8y4 EcgTuDQQ7pXJA2d55JxQO8uZTGZvGTuhJOgeE=*In-reply-to*: <175241C47FF8E141A7AC2637D952A66D0A5FBA0567@ITUPC-EX1MBOX.UniNet.unisa.edu.au>*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*: <00dd01c9fefa$6dc53690$8b55d383@w2k3.fi.uu.nl> <2C7F9BCA-5807-42F9-8AB7-CC1F7EE9F9FE@umich.edu> <4A553223.8020809@mira.net> <30364f990907082057i3af7ecf4w93fc2bcc8b99cb8c@mail.gmail.com> <000601ca0631$0bd038c0$0202a8c0@youre6f02835ae> <D322A7F4-2366-4A92-8484-D876BDB272BF@ucsd.edu> <14a6419f0907201907u751a9eddm11cf5c50df9075da@mail.gmail.com> <175241C47FF8E141A7AC2637D952A66D0A5FBA0567@ITUPC-EX1MBOX.UniNet.unisa.edu.au>*Reply-to*: mcole@weber.ucsd.edu, "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Sender*: xmca-bounces@weber.ucsd.edu

I also find the "marching on the number line" idea helpful, Monica. There ought to be several on-line ways to embody this way of thinking about the problem, but I have not found them. I personally, intuitively, find the "a minus times a plus" easier to understand than a minus times a minus". For the latter, something like the idea of the sign as an "operator" seems to be required. But I'll bet your way of doing things could be a great outdoor game for kids. How do you do multiplication? mike On Tue, Jul 21, 2009 at 1:23 AM, Monica Behrend <Monica.Behrend@unisa.edu.au > wrote: > Hi, having been an XCMA lurker and a post maths teacher and maths educator, > I have followed this discussion with interest - and certainly concur with > the conceptual issues around this! > > Two analogies that have somehow promoted Eureka moments for my students and > student teachers have been: > > 1. Marching forwards (+ add) and backwards (- subtract) along number lines. > We reckoned that we defined the +4 as walking four steps in the direction of > the larger numbers, and -4 as 'turn around' and walk four steps! We had > great fun and I still remember a trainee secondary teacher from a Pacific > Island country doing that with much aplomb! That can be extended to the > multiplication. Somehow though then we wanted to not step but jump in lots > of 4! So we turned into frogs! > > 2. Money ... so -4 X 4 = four dollar debt by 4 ... means you owe 16 dollars > i.e. -16. So -4 x -4 means four dollar debt and four times repaid ... +16. > This one is not so neat, but can make the point too. > > Cheers > > Monica Behrend > Research Education Adviser (International) > The University of South Australia > Adelaide, SA Australia > > > > > -----Original Message----- > From: xmca-bounces@weber.ucsd.edu [mailto:xmca-bounces@weber.ucsd.edu] On > Behalf Of Ng Foo Keong > Sent: Tuesday, 21 July 2009 11:38 AM > To: eXtended Mind, Culture, Activity > Subject: Re: [xmca] a minus times a plus > > from my teaching experience, the main confusion is between (in our > conventional > notation) -4 x -4 = 16 and -4 - 4 = -8. > > learners must know the difference in meanings between the two. > > in mathematics, the definitions of operations and the use of symbols > are actually > arbitrary, and it is possible to explore alternative systems of > algebra (like what > professional mathematicians / algebraists do) as long as they make sense > and are > consistent. > > i'd say, just like non-Euclidean geometries, alternative algebras would be > interesting topic for exploration away from the boredom of "normal math", > but students still need to be able to discern different meanings and use > the appropriate different symbols. the systems that students > propose/explore > should make sense, be well-defined and be consistent (non-contradictory) -- > what > the community of mathematicians would require. explorations give > students a sense > of agency and creativity and builds their identity as active contributors > of > mathematical knowledge (instead of being passive followers of algorithms). > > F.K. > > > > > 2009/7/20 Jerry Balzano <gjbalzano@ucsd.edu>: > > Hi Mike, from across UCSD campus ... actually from across the country > since > > I'm currently in NY ... > > > > by my count, this topic has accumulated 147 emails since your original > April > > 27 posting (this one would be #148) ... quite a fecund topic, and not bad > on > > the longevity meter either! (nearly 3 months) > > > > I just this morning ran across another remarkable connection to the topic > > that I had to tell you and everyone else about as I was in google > bookland, > > pursuing cross refs to -- of all things -- WIttgenstein's Lectures on the > > Foundations of Mathematics. It's a rather fascinating book called > Negative > > Math, by Alberto A. Martínez, and the online "book overview" starts off, > > believe it or not, just like this: > > > > A student in class asks the math teacher: "Shouldn't minus times minus > make > > minus?" > > > > There's a chapter in the book with the seemingly heretical title, "Math > is > > Rather Flexible", and as if to demonstrate this via a kind of tour de > force > > with an exceptional resonance for our discussion, Martinez asks "can we > > construct a system in which, say, -4 x -4 = -16? Actually, yes we can." > > > > This raises the question: Is such a book good for students or bad for > > students? It seems terribly subversive, doesn't it? I can imagine more > > than one math teacher applauding it "in principle" but panning it in > > practice for fear that it "might confuse" a student who was "having > enough > > trouble learning the (correct) rules". But (on the other hand) perhaps > if > > we had a more playful, less rigid attitude about "the rules", we would > > engender a less fearful attitude in students about them. Perhaps books > in > > the spirit of Martinez' Negative Math would be a proper antidote to such > > (apparently!) unproductive approaches to thinking about teaching and > > learning mathematics? > > > > The book is in our library at UCSD, and I'd be more than happy to "play > with > > it" with you when I return (beginning of August), if you like. In the > > meantime, the Google Books link is here. > > > > Jerry B > > > > ------- > > > > > > On Jul 16, 2009, at 9:18 AM, Esther Goody wrote: > > > >> Dear Mike, > >> Hope you have caught up with sleep since Alaska! > >> Until now I have not looked at the "a minus times a plus" topic in XMCA, > >> supposing it would be about word games or something. Now I see it is > about > >> 'How and What to teach in school maths'? This is something I stumbled > into > >> in my northern Ghana classrooms. The first Spencer Foundation grant was > >> about differences in learning literacy skills in L1 and L2 in high and > low > >> authority classrooms. However a large section of the middle year report > >> was > >> about reasons why kids were not learning school maths in the upper > primary > >> grades in village schools. > >> > > ...... > > . > > . > > . > > . > > . > > . > > . > > . > > . > > . > > > > _______________________________________________ > > 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 > _______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca

**References**:**RE: [xmca] a minus times a plus***From:*"A.Bakker" <A.Bakker@fi.uu.nl>

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

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

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

**RE: [xmca] a minus times a plus***From:*"Esther Goody" <eg100@hermes.cam.ac.uk>

**Re: [xmca] a minus times a plus***From:*Jerry Balzano <gjbalzano@ucsd.edu>

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

**RE: [xmca] a minus times a plus***From:*Monica Behrend <Monica.Behrend@unisa.edu.au>

- Prev by Date:
**Re: [xmca] Half a coconut** - Next by Date:
**RE: [xmca] http://www.elluminate.com/support/** - Previous by thread:
**RE: [xmca] a minus times a plus** - Next by thread:
**RE: [xmca] a minus times a plus** - Index(es):