Re: [xmca] a minus times a plus

[Mike Cole <lchcmike@gmail.com>]

Andy-- i am just back from Alaska and want to talk about several matters,
but am falling down tired.
Re the problem under discussion. You have now encountered Luria and
context-variable learning AND DEVELOPMENT
from another angle. I am waaaaay behind the 8 ball but will try to be on
skype with you asap.
mike

On Wed, Jul 8, 2009 at 4:56 PM, Andy Blunden <ablunden@mira.net> wrote:

> Thanks for that Jay. You make your points very strongly. But if it is true
> that for the learning of mathematics:
>
> "almost all practices and procedures are highly context and content
> specific, with what appears otherwise resulting mostly from certain sorts of
> cultural habitus acquired by individuals. Classes of tasks and practices
> have historically developed with certain kinds of similarities within
> particular cultures. Cultures have preferred strategies or ways-of-doing
> that are implicit, and members who have the appropriate trajectories of
> cultural experiences tend to develop dispositions fitted to these."
>
> then this does not negate the value of lerning mathematics, does it? It
> merely suggests that there may be other ways of achieving the same end. I am
> guessing that participating in formal debate teaches similar dispositions.
> Does this come up against the same obstacles? I rather suspect it does, but
> of course I have no evidence.
>
> Isn't acquiring a fitting disposition effectively the same as
> context-variable learning?
>
> Andy
>
>
> Jay Lemke wrote:
>
>> I'm pleased there has been uptake from Ivan's and my concerns about school
>> mathematics, and science.
>>
>> A few miscellaneous responses --
>>
>> I don't doubt that there are many, even semi-technical, occupations where
>> mathematical procedures are employed, and even where (though I suspect much
>> less often) some judgment is required about how to apply them (thus
>> requiring at least some theoretical understanding). I will be interested to
>> see what Bakker's research shows about the most prevalent of these -- how
>> advanced a level is used so widely that it is efficient to teach it to ALL
>> students in schools? I really don't imagine very many people factor
>> polynomials or solve quadratic equations, outside of higher level
>> specializations. But these are empirical questions, whereas the content of
>> the curriculum is NOT based on empirical findings of this sort, but rather
>> on traditions of dubious validity.
>>
>> How to teach the mathematics that is widely used is then a separate
>> question, and I think there is growing agreement that more realistic
>> contexts are better for gaining wider success. There is still the very
>> fundamental issue of whether translating such contexts into school
>> activities can work well and generally (which I tend to doubt), or whether
>> the learning needs to be taken outside of classrooms, or at least into mixed
>> settings that combine experience and experience-based intuitions from
>> non-school settings with some reflection and analysis work in classrooms,
>> etc. Obviously SOME math can be taught successfully in classrooms, like some
>> literacy skills, and some translation strategies are of value.
>>
>> But I would agree that the abstract approach to math and science, and the
>> overstuffed topic curricula in these fields, is there more because (a) we
>> know how to segment it and test it, and (b) it's a good way to keep a lot of
>> people out of universities and professional jobs, while seeming to be
>> completely objective and fair about a rigged system.
>>
>> Does it benefit the mind in more general ways? I am a splitter ... I don't
>> believe much in transfer, generalization, general intelligence,
>> multi-purpose higher mental functions, etc. I tend to think that almost all
>> practices and procedures are highly context and content -specific, with what
>> appears otherwise resulting mostly from certain sorts of cultural habitus
>> acquired by individuals. Classes of tasks and practices have historically
>> developed with certain kinds of similarities within particular cultures.
>> Cultures have preferred strategies or ways-of-doing that are implicit, and
>> members who have the appropriate trajectories of cultural experiences tend
>> to develop dispositions fitted to these.
>>
>> In these terms, experiences with mathematics CAN support developing
>> dispositions that make mastering other kinds of abstract reasoning practices
>> come more easily. Symmetrically, mastering mathematics is easier if you've
>> already had success with other implicitly similar kinds of tasks and
>> strategies. Learning abstract decontextualized mathematics, however, seems
>> to me one of the hardest ways into such a cultural complex of similar
>> practices. And any benefit from working at mathematics seems to me to accrue
>> only if (a) the work is enjoyable or at least has a supportive relationship
>> to a desired identity, and (b) you are successful at it, preferably early
>> on.
>>
>> All this applies to conceptual understanding of sciences equally as well.
>>
>> JAY.
>>
>> Jay Lemke
>> Professor
>> Educational Studies
>> University of Michigan
>> Ann Arbor, MI 48109
>> www.umich.edu/~jaylemke <http://www.umich.edu/%7Ejaylemke>
>>
>>
>>
>>
>> On Jul 7, 2009, at 1:59 PM, A.Bakker wrote:
>>
>>  Interesting discussion! Let me dwell on two projects in response to Jay
>>> and
>>> Andy.
>>>
>>> 1. what kind of math do we need at work?
>>>
>>> We have analyzed the mathematical knowledge required in 239
>>> intermediate-level professions (think of service engineering, florist,
>>> baker, low level analyst in science labs, builders, car mechanics, salary
>>> administration, secretarial work, hairdresser etc). Some of these do not
>>> have to do any calculations at all (butcher in a factory just selecting
>>> good
>>> and bad parts of meat), but the vast majority of professions face simple
>>> arithmetic, geometry (area, volume), data handling and risk, and
>>> sometimes
>>> formulas. Even at the lowest level of education, lab analysts face some
>>> high-level statistics (F-test, t-test, correlation etc) in method
>>> validation, precision of instruments etc.
>>>
>>> Although there is some truth in Andy's comments, this analysis gives a
>>> more
>>> nuanced image. Moreover, there is more than math at work and in daily
>>> life:
>>> math required for higher-level education. Vocational students without
>>> enough
>>> mathematical and scientific baggage have trouble getting through their
>>> higher vocational education (nursing, teaching, management etc). however,
>>> I
>>> should note that our Dutch school system differs drastically from the
>>> American one because our vocational education is big (60% of the
>>> students)
>>> and starts early (pre-vocational education at age 12).
>>>
>>> 2. basing science units on authentic practices
>>>
>>> Indeed, many math and science problems at school are not very realistic.
>>> It
>>> is in fact quite hard to design good ones. Over the past years we have
>>> tried
>>> to base educational units on authentic practices in which science or math
>>> is
>>> used (with activity theory in mind as well). We have 'translated'
>>> authentic
>>> goals to learner goals, adapted ways of working and knowledge required to
>>> be
>>> manageable to students (grade 10-12). The idea was to use meaningful
>>> relationships between goals, tools, knowledge etc in outside-school
>>> practice
>>> as sources of inspiration for school units. Although we have had some
>>> success, there are still many challenges in designing good units - even
>>> if
>>> we allow the learning goals to be drastically different from the
>>> Standards
>>> (say: insight into health and nutrition rather than say DNA, evolution,
>>> cell
>>> biology...).
>>>
>>> So I agree with Jay that content is a major problem, but even then we
>>> have a
>>> lot of work to do in terms of designing good alternatives.
>>>
>>> Chevallard has written interesting papers on didactic transposition,
>>> adapting knowledge as used in the 'real world' to school situations. He
>>> describes a contradiction that cannot be resolved completely: education
>>> promises to prepare kids for their future and for society. At the same
>>> time,
>>> education cannot really fulfil its promise. What students learn is often
>>> something that teachers can easily test. Chevallard argues that the main
>>> reason that we still teach math and science is NOT that they are so
>>> useful,
>>> but they can be rolled out nicely in stages over the school grades and
>>> can
>>> be tested in objective ways. A lot of things that are very useful to
>>> learn
>>> do not make it to the curricula, simply because they are so hard to teach
>>> and test (medicine, psychology, sociology, social skills etc.)
>>>
>>> Arthur Bakker
>>>
>>>  -----Original Message-----
>>>> From: xmca-bounces@weber.ucsd.edu [mailto:xmca-bounces@weber.ucsd.edu]
>>>> On
>>>> Behalf Of Andy Blunden
>>>> Sent: dinsdag 7 juli 2009 13:30
>>>> To: eXtended Mind, Culture, Activity
>>>> Subject: Re: [xmca] a minus times a plus
>>>>
>>>> Your key claims are beyond challenge Jay; you can get by
>>>> perfectly well in all aspects of life without mathematics,
>>>> apart from a basic understanding of the notion of quantity
>>>> and some elementary arithmetic, except for a very small
>>>> group of professions. It has annoyed me, this need to invent
>>>> pseudo-problems that seem to demand mathematics, to
>>>> "justify" the need to learn maths. It seems to me that it is
>>>> requirement to pass maths exams to gain entry to a very wide
>>>> range of jobs etc., which is the only real motivation for most.
>>>>
>>>> But can you tell me, is there no evidence that going through
>>>> the process of learning maths in some way benefits the mind?
>>>> in the same way that (as I understand it) learning to read
>>>> and write has a permanent and effect on how people think?
>>>> that mathematics is a kind of mental gymnastics.
>>>>
>>>> Andy
>>>>
>>>> Jay Lemke wrote:
>>>>
>>>>> Thank, Ivan, for responding in part to some of my concerns re teaching
>>>>> math-as-math in schools.
>>>>>
>>>>> It's a big, old debate in education whether we should teach ideas,
>>>>> concepts, and disciplines as abstract systems, in the hopes they can
>>>>> then be used as tools to think with ... or whether that usually doesn't
>>>>> work, puts kids off from the subject, and it's better to let concepts
>>>>> appear more naturally in the context of real-world problems, issues,
>>>>> activities which are not about math or science, but in which math-using
>>>>> and science-using activities and practices can play a helpful part.
>>>>>
>>>>> The academic, and intellectual answer, as part of a cultural and
>>>>> institutional tradition, is that we cheat students out of the power of
>>>>> math and science if we don't give them the systems of abstract
>>>>> concepts,
>>>>> and that other approaches tend to degenerate into second-rate
>>>>> practicalism that avoids theory, critique, alternatives, creativity,
>>>>> etc. My own view, after a long time participating in, observing, and
>>>>> trying to analyze the teaching of science, and to a lesser degree
>>>>> mathematics, is that the powerful systematic conceptual tools are a
>>>>> very
>>>>> advanced stage of membership in one or more very specialized
>>>>> communities, and are simply not of much use to most people.
>>>>>
>>>>> Maybe my view is a bit extreme. But I think it remains true that it is
>>>>> not just a failure to find the magic method of teaching that is the
>>>>> problem with math-as-math and science-as-science in the curriculum. It
>>>>> is the content itself. Or, really, the lack of content, the lack of
>>>>> engagement with real life activities that are meaningful and important
>>>>> to the students, in the modern math and science curricula. And I do not
>>>>> see the solution as inventing clever artificial problems and topics
>>>>> that
>>>>> seem to be relevant to real-life, but which are in fact just excuses to
>>>>> do more math-as-math and science-as-science.
>>>>>
>>>>> A mathematician or a scientist can find, show you, highlight, apply
>>>>> their conceptual tools to nearly anything. Some reasonable level of
>>>>> abstract awareness of those tools can emerge from encountering, in some
>>>>> detail and depth, several domains and examples or projects in which the
>>>>> concepts have been highlighted for their usefulness (and that includes
>>>>> usefulness for critical thinking, for imagining alternatives --- not
>>>>> just for engineering practical constructions or solutions). But this
>>>>> comes at the end of a long learning process, and almost as a kind of
>>>>> side-effect, and not at the beginning or as the primary purpose or
>>>>> goal-of-activity.
>>>>>
>>>>> There is math and science in jumping jacks and football, in mountain
>>>>> climbing, in raising a pet or growing some food, in figuring the cost
>>>>> of
>>>>> better garbage collection in the neighborhood, in organizing a block
>>>>> party, in understanding when to go to the hospital or what counts as
>>>>> evidence in a court case. It might be better to say that there are
>>>>> issues of quantity and degree, of probability and risk, of nutrition
>>>>> and
>>>>> cause and effect in all these domains and phenonena, and that the
>>>>> workarounds and tricks and mnemonics and practical methods accumulated
>>>>> across them all tend to implicate some more general strategies ---
>>>>> which
>>>>> we could just tell you, but then the odds are you wouldn't understand
>>>>> or
>>>>> remember or know how to use them for yourself.
>>>>>
>>>>> I am not talking here about advanced levels of education, but about
>>>>> introductory ones ... up to about the age of 15 or 17, or up to the
>>>>> point at which interest and possibility tend to focus students toward
>>>>> some preferred specialization. Then the balance shifts, again not all
>>>>> the way toward abstract disciplines (as, for example, medical education
>>>>> has struggled to sort out for a long time now), but a bit more toward
>>>>> the justification of more emphasis on theoretical learning, as part of
>>>>> membership in a specialist community of knowers/doers.
>>>>>
>>>>> What are the practical situations in which you need to multiply a minus
>>>>> times a plus? not textbook imaginaries, but for real? If you had some
>>>>> broad and in-depth knowledge about such a situation, would it then be
>>>>> so
>>>>> hard to make sense of how signed numbers multiply there? And how far a
>>>>> step is it, and how necessary a step for all to take, from an induction
>>>>> from several such well-understood situations to the pure
>>>>> mathematicians'
>>>>> abstract arguments about how signed numbers multiply everywhere, or
>>>>> really, nowhere??
>>>>>
>>>>> JAY.
>>>>>
>>>>>
>>>>>
>>>>> Jay Lemke
>>>>> Professor
>>>>> Educational Studies
>>>>> University of Michigan
>>>>> Ann Arbor, MI 48109
>>>>> www.umich.edu/~jaylemke <http://www.umich.edu/%7Ejaylemke>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> On Jun 30, 2009, at 6:50 PM, Ivan Rosero wrote:
>>>>>
>>>>>  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
>>>>
>>>>> Constantine.
>>>>>>
>>>>>> 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?
>>>>>>
>>>>>> Ivan
>>>>>>
>>>>>> 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 <http://www.umich.edu/%7Ejaylemke>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> 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
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> _______________________________________________
>>>>>>>>> xmca mailing list
>>>>>>>>> xmca@weber.ucsd.edu
>>>>>>>>> http://dss.ucsd.edu/mailman/listinfo/xmca
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> _______________________________________________
>>>>>>>>>
>>>>>>>> xmca mailing list
>>>>>>>> xmca@weber.ucsd.edu
>>>>>>>> http://dss.ucsd.edu/mailman/listinfo/xmca
>>>>>>>>
>>>>>>>>
>>>>>>>>  --
>>>>>>> ----------------------------------------------------------------------
>>>>>>>
>>>>>>>
>>>>>> --
>>>>
>>>>> Andy Blunden (Erythrós Press and Media) http://www.erythrospress.com/
>>>>>>> Orders: http://www.erythrospress.com/store/main.html#books
>>>>>>>
>>>>>>> _______________________________________________
>>>>>>> xmca mailing list
>>>>>>> xmca@weber.ucsd.edu
>>>>>>> http://dss.ucsd.edu/mailman/listinfo/xmca
>>>>>>>
>>>>>>>  _______________________________________________
>>>>>> xmca mailing list
>>>>>> xmca@weber.ucsd.edu
>>>>>> http://dss.ucsd.edu/mailman/listinfo/xmca
>>>>>>
>>>>>>
>>>>>>
>>>>> _______________________________________________
>>>>> xmca mailing list
>>>>> xmca@weber.ucsd.edu
>>>>> http://dss.ucsd.edu/mailman/listinfo/xmca
>>>>>
>>>>>
>>>> --
>>>> ------------------------------------------------------------------------
>>>> Andy Blunden (Erythrós Press and Media)
>>>> http://www.erythrospress.com/
>>>> Orders: http://www.erythrospress.com/store/main.html#books
>>>>
>>>> _______________________________________________
>>>> xmca mailing list
>>>> xmca@weber.ucsd.edu
>>>> http://dss.ucsd.edu/mailman/listinfo/xmca
>>>>
>>>
>>> _______________________________________________
>>> xmca mailing list
>>> xmca@weber.ucsd.edu
>>> http://dss.ucsd.edu/mailman/listinfo/xmca
>>>
>>>
>>>
>> _______________________________________________
>> xmca mailing list
>> xmca@weber.ucsd.edu
>> http://dss.ucsd.edu/mailman/listinfo/xmca
>>
>>
> --
> ------------------------------------------------------------------------
>
> Andy Blunden (Erythrós Press and Media) http://www.erythrospress.com/
> Orders: http://www.erythrospress.com/store/main.html#books
>
> _______________________________________________
> xmca mailing list
> xmca@weber.ucsd.edu
> http://dss.ucsd.edu/mailman/listinfo/xmca
>
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