# Re: [xmca] a minus times a plus

```hi all,

(1) i am happy to see Anna Sfard in this mailing list, as i
"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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```