Re: [xmca] a minus times a plus

[Ng Foo Keong <lefouque@gmail.com>]

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
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