Yes, Mike and F.K., these are very disturbing issues. Both that what we
think we want to teach seems to depend on deeper (e.g. 4000-year deep)
knowledge than it's realistic to expect most people to learn (or want to
learn), and that how we teach even the most practical bits of mathematics
(like 15 minus 8) seems to have gone so wrong that it's hard to know where
to start, especially for those we have most systematically failed.
We do indeed need to not give up. But we also need, I think, to admit that
it's time to seriously re-think the whole of the what, why, and how of
education. Math is a nice place to focus because at least some of it seems
universally agreed to be useful by almost everyone, because professional
mathematicians and most people, including teachers and mathematics
educators, seem to hold radically different views about what the subject is,
and because success in teaching it, measured in almost any way, is pretty
near the bottom of the heap.
Yes, we can find somewhat better ways to teach the same stuff, but maybe
it's the stuff itself (the content of the curriculum, viewed not just as
information, but as activity) that needs to be rethought? along with the
ethics and efficacy of who decides.
No matter how many times you multiply a minus by any number of pluses, you
still get a minus.
University of Michigan
Ann Arbor, MI 48109
On Jun 6, 2009, at 6:12 PM, Mike Cole wrote:
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 <firstname.lastname@example.org> 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
thus the huge gulf of understanding still persists. that's why
as an educator, i feel so useless being unable to help other
2009/6/4 Mike Cole <email@example.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
me. Now that I am done teaching and have a whole day to communicate
are looking up!! Apologies for doubting I could have deep understanding
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
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.
And member book links are coming in. Nice.
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