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Re: [xmca] a minus times a plus

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 source
is quite practical with social significance: the unbridgable gap the
children 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.

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
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