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Re: [xmca] a minus times a plus

Here's a familiar exhortation:
"We need as many engineers as possible.  As there is a lack of them, invite
to this study, persons of about 18 years, who have already studied the
necessary sciences. Relieve the parents of taxes and grant the scholars
sufficient means."

According to my brief cyber-sphere search, these are the words of Emperor

So, anyway, we all know what road that empire took.  I doubt it was lack of
engineers though :)  So, given the very similar verbiage spilling out of NSF
these days, I agree with Jay, perhaps slowing down and taking a minute or
two to rethink this wouldn't be bad at all.

If I read you correctly Jay, one big worry you have is that we don't end up
reifying mathematics (in the sense Constantine seems to be doing with
engineering) in the frustration we experience with our almost complete
failure in teaching it.

It reminds me of mountain-climbing.  For me at least, this is one hell of a
difficult sport, and the few times I've ever participated, it has been a
real big struggle to get to the top.  And we're talking Mt. Washington, a
measly ~6000 ft peek.  Anyway, I struggle, sweat, almost pass-out, and
finally I'm there.  It is AWESOME, the joy is overwhelming.  20 minutes
later, as my muscles cool down and my adrenaline levels-off, I stare down
the thing and feel a creeping dread, even if the way down is many times
easier than the way up.

This story can go in many directions from here, as many as there are people
who have made it (oh, God, this is cheesy) mountain-top.  They are not
universally happy stories however.

I DO think it is useful to know some mathematics and have a host of
scientific concepts to think with and through at our disposal.  None of this
is Bad (or Good for that matter) in and of itself.  The Purpose, of course,
is what is at issue.

ZPDs are value agnostic.  Mike and his team at LCHC are currently attempting
to create ZPDs that can instill basic arithmetic in kids whose daily (and
arguably far stronger) ZPDs pull them in many other (sometimes directly
opposite) directions.  Some of those ZPDs, however, are not in direct
conflict with math.  That is my hunch, or assumption.  The task, then, is
perhaps a bit simpler than creating new ones.

Is it simpler to find and then piggy-back on, ZPDs that contain kernels of
arithmetic in them?  Susan Goldin-Meadow has pretty convincing evidence that
specific motor activity can not only presage basic arithmetic, but can even
aid in its acquisition.  So, might not Jay's concern (if I read him right)
that mathematics (and the whole lot of techno-science) becomes
surreptitiously reified in our frustrated attempts to teach it be addressed
from a different direction?

Jumping-jacks anyone?


On Sat, Jun 27, 2009 at 11:00 PM, Andy Blunden <ablunden@mira.net> wrote:

> I hope people won't mind if I continue to pick the brains of this list on
> the problem of my niece's progress in maths, or lack of it.
> It seems that the suggestion last time - that Marissa may have missed
> important lessons while on holiday - may explain her poor performance last
> year in maths, even though maths has always been her weak subject. She has
> caught up a bit but she is still badly behind.
> It seems that the issue Mike has raised also applies: she is getting
> homework that seem to presume she know things that in fact she doesn't. The
> only other negative in her school reports is that she doesn't participate in
> class discussion or ask questions when she doesn't understand something.
> I presume the hesitancy about speaking up is probably the cause of failure
> to correct her maths problems and the teachers giving her homework she
> doesn't understand.
> She is now 15 and her maths homework is also beyond her father! :) and the
> crisis of the transition from childhood to adulthood around this age, makes
> it impossible for the father to get Marissa talk about it to him, or engage
> Marissa in games of 20 Questions or something to lead her to the joys of
> asking others. Discussion over the dinner table is apparently also
> unconducive to her participation.
> Does anyone have any ideas? I've run out of suggestions. I could probably
> help if I was there, but I'm 1000 km away.
> Andy
> Mike Cole wrote:
>> SO glad you are interested in this, Jay.
>> I have just made contact with Karen Fuson who has, lucky for us, "retired"
>> from Northwestern and moved nearby. She is away for a week or so
>> but then we are getting together. This is a problem that just may be
>> tractable, theoretically interesting for sure, attractive of experience
>> collaborators,
>> and god knows, of practrical importance to lots of kids.
>> mike
>> On Sun, Jun 7, 2009 at 3:27 PM, Jay Lemke <jaylemke@umich.edu> wrote:
>>  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.
>>> JAY.
>>>  Jay Lemke
>>> Professor
>>> Educational Studies
>>> University of Michigan
>>> Ann Arbor, MI 48109
>>> www.umich.edu/~jaylemke
>>>  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.
>>> mike
>>> 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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> --
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