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*To*: ablunden@mira.net, "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Subject*: Re: [xmca] a minus times a plus*From*: Ng Foo Keong <lefouque@gmail.com>*Date*: Wed, 29 Apr 2009 00:38:53 +0800*Cc*:*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:in-reply-to:references :date:message-id:subject:from:to:content-type :content-transfer-encoding; bh=LDXP540DcYaoj2LN6KBrGmipLTaso0yzS5le37zu4ns=; b=AW44tbyeRgilenXvBdBeDJMx84yYItmaZDLCB5DFSf/Xo2UO1IS4xuEeIZoCnkFrLg ljJIkz8FSP2Tm4SFI43hiXF3mc6Iguu1jnl1q1Sp/LDpi7rTOqnB3DlmILxfR3iSO1cf tErKzJ0T9tu/l4DdEC/9hjrYG5UrgqTUVXJ6o=*Domainkey-signature*: a=rsa-sha1; c=nofws; d=gmail.com; s=gamma; h=mime-version:in-reply-to:references:date:message-id:subject:from:to :content-type:content-transfer-encoding; b=fo+RYBkQfSQBia7fyJVWNRD/+aYZAtqtaN24BIVJHHXejrtIk5sRyws+To9mK/eTHP MRY0QadKm5pmUUlTiFwah5A97y4m1lQrfXgHHOkSCXWoMk9pzkVg03yVecjZOWwnRRfh su7g3rboXt5LdZ6+f4aOXWzqedad8SQ80ZjUY=*In-reply-to*: <49F6DC22.4060002@mira.net>*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*: <30364f990904271547o5b4df21eifca69bf8318483f2@mail.gmail.com> <0193A85F-03A7-4811-B912-217722A5770B@uvic.ca> <C23F99F4-D4AC-463B-AF11-30585E74F7CF@me.com> <30364f990904271949r3cfef116m89d39ea419d112b2@mail.gmail.com> <p06240805c61c2a15a97a@192.168.1.65> <49F68AEE.40502@mira.net> <3B19033D3E2EC34C97DF364119A79A61C90CDD@EXVS1.its.uidaho.edu> <49F69438.4050900@mira.net> <18289BE6B2A646DFABDC1C342E7ED5A1@x2a4f0b4148df4> <49F6DC22.4060002@mira.net>*Reply-to*: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Sender*: xmca-bounces@weber.ucsd.edu

hi all, (1) i am happy to see Anna Sfard in this mailing list, as i am adopting/adapting her framework for 'identities' as "reified, significant and endorsable" life stories in my PhD study of socio-mathematical identities of trainee teachers. (2) i agree with Jerry Balzano that maths is about patterns. (numbers are just one kind of pattern among many -- the kind that make you "numb"). However, the mirror analogy is not so "pre-numerical", upon deeper ... erm ... reflection: if matrices A and B represent reflections in 3D space represented respectively by matrices A and B (both of their determinants are -ve) then the composite transformation represented by the matrix AB will have +ve determinant det(AB) = det(A) det(B), well in accord with our rule of signs. i guess we're really trying to teach reasoning by analogy or by metaphor. professional mathematicians use fancier terms like "homomorphisms" and "functors". (3) IMHO math students should learn think in multiple ways and see the various explanations are 'homomorphic' in some way. my favorite one is the determinant one, but maybe its for those who already 'understand / got used to" it. i like the ice-cubes one too. For our troubled financial times, i offer the following analogy:- e.g. -2 × -3 can be interpreted as removing/cancelling two times debts of $3. this is the same as 2 × 3 i.e. paying back two times $3. (4) my 2 cents' regarding "rule change" and "participation":- it would appear that math teachers and students still need to include, as part of their discourse, similarities and differences between what they are learning and what they have learnt, even if they are not using professional jargon like "construction via ordered-pairs", "the ring of integers", "natural embedding", "closure with respect to subtraction", ... etc. If "properly done", their language game should asymptotically tend towards isomorphism with the professionals' -- a 'mathematics' by another name. ;-P Foo Keong NIE, Singapore _______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca

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

**Re: [xmca] a minus times a plus***From:*Wolff-Michael Roth <mroth@uvic.ca>

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

**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:*"Duvall, Emily" <emily@uidaho.edu>

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

**Re: [xmca] a minus times a plus***From:*Anna Sfard <annasfar@math.msu.edu>

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

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