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RE: [xmca] a minus times a plus

Dear Mike,
	Hope you have caught up with sleep since Alaska!
Until now I have not looked at the "a minus times a plus" topic in XMCA,
supposing it would be about word games or something. Now I see it is about
'How and What to teach in school maths'? This is something I stumbled into
in my northern Ghana classrooms. The first Spencer Foundation grant was
about differences in learning literacy skills in L1 and L2 in high and low
authority classrooms. However a large section of the middle year report was
about reasons why kids were not learning school maths in the upper primary
grades in village schools. 
	This was clearly different from the problem of not learning to read,
since reading was being taught only in English/L2 to village children who
never even heard this language. 'Solving' the reading skills problem only
required teaching kids to read first in L1. While observing a good
government teacher taking the upper classes through maths lessons on 'profit
and loss' (in English, of course,) it was evident that the kids did not
understand the meaning of what the teacher said. Basically they couldn't
follow the problems, and even where they did, profit and loss are not made
explicit in local calculations. Yet these were kids who are quicker than I
am in local market transactions - which are based on making at least a small
profit in each instance. But local market trading goes on in L1 of course.
There are several maths routines which are combined to do complex
calculations quickly. (The Brazilian work on street maths found local
routines critical.)
	In order to open up what was going on so I could follow, I started a
maths club after school and on Saturdays. (Thank you Mike for the idea and
model.) There were about nine kids in the upper three grades. Five or six
came regularly. I got them to tell me about what they did outside school
that involved making and selling something. Half the girls helped a
mother/aunt/older 'sister' with brewing beer in their compound to sell to
villagers who drop in to drink it there. One girl fried groundnut paste to
sell as a snack on the weekly market day; she did this with/for an older
'sister'. One boy salvaged torn bicycle inner tubes and made slingshots to
sell to his mates. Two or three boys reported working with others on their
fathers' farms (It proved difficult to glean much for profit/loss from farm
work, since this involves long-term reciprocity.) 
	For three of these activities in turn, beer-brewing, frying
groundnut snacks and making slingshots) we sat together and worked out
costs, and income. Then I would take the class, and lead the kids to figure
out what were costs and income. Finally I would try to put our 'findings'
into the formal 'profit/loss' format used in maths problems. Except for the
slingshots, it wasn't really possible to arrive at profit.  
	Pondering this afterwards, it seemed to me that in effect there was
an M1 in the village in which children become skilled by participating in
everyday activities. Formal school maths (M2)ignores this, and as children
'learn' maths 2 they do not relate it to solving everyday problems they
already do easily in M1. When I mentioned this recently to a USAID
consultant who usually works in rural Egypt, he agreed strongly. We both
think the same thing probably happens in US and UK schools where a single
language is involved.
	Is any of this relevant? Worth pursuing?

Can you give me more to go on for the references about dialogue involving
ROTH? Does he have a first name. Can you recall any titles?
		Cheers, Esther

-----Original Message-----
From: xmca-bounces@weber.ucsd.edu [mailto:xmca-bounces@weber.ucsd.edu] On
Behalf Of Mike Cole
Sent: 09 July 2009 04:57
To: ablunden@mira.net; eXtended Mind, Culture, Activity
Subject: Re: [xmca] a minus times a plus

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.

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
> 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
> guessing that participating in formal debate teaches similar dispositions.
> Does this come up against the same obstacles? I rather suspect it does,
> 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
>> 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
>> less often) some judgment is required about how to apply them (thus
>> requiring at least some theoretical understanding). I will be interested
>> 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
>> 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
>> the curriculum is NOT based on empirical findings of this sort, but
>> 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
>> the learning needs to be taken outside of classrooms, or at least into
>> 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
>> 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
>> 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
>> believe much in transfer, generalization, general intelligence,
>> multi-purpose higher mental functions, etc. I tend to think that almost
>> practices and procedures are highly context and content -specific, with
>> 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,
>> members who have the appropriate trajectories of cultural experiences
>> to develop dispositions fitted to these.
>> In these terms, experiences with mathematics CAN support developing
>> dispositions that make mastering other kinds of abstract reasoning
>> come more easily. Symmetrically, mastering mathematics is easier if
>> already had success with other implicitly similar kinds of tasks and
>> strategies. Learning abstract decontextualized mathematics, however,
>> 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
>> only if (a) the work is enjoyable or at least has a supportive
>> 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,
>>> 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).
>>> 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
>>> is
>>> used (with activity theory in mind as well). We have 'translated'
>>> authentic
>>> goals to learner goals, adapted ways of working and knowledge required
>>> 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
>>> 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
>>>>> 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
>>>>> 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
>>>>> 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
>>>>> 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
>>>>> detail and depth, several domains and examples or projects in which
>>>>> 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
>>>>> 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
>>>>> 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
>>>>> 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
>>>>>> 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
>>>>> 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
>>>>>> failure in teaching it.
>>>>>> It reminds me of mountain-climbing.  For me at least, this is one
>>>>>> of a
>>>>>> difficult sport, and the few times I've ever participated, it has
>>>>> a
>>>>> real big struggle to get to the top.  And we're talking Mt.
>>>>> 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
>>>>>> 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
>>>>>> 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
>>>>>> opposite) directions.  Some of those ZPDs, however, are not in direct
>>>>>> conflict with math.  That is my hunch, or assumption.  The task,
>>>>> 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
>>>>>> 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
>>>>>>> important lessons while on holiday - may explain her poor
>>>>>>> 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
>>>>>>> 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
>>>>>> 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
>>>>>>>> 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
>>>>>>> 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
>>>>>>>> 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!
>>>>>>>>> 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
>>>>>>>>> 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
>>>>>>>>> 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
>>>>>>>>> the
>>>>>>>>> teacher is focused on is
>>>>>>>>> disabled because when asked 15-8 the answer =3 and only
>>>>>>>>> attention to the problem set up with fingers and subtracting one
>>>>>>>>> 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
>>>>>>>>> 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
>>>>>>>>> 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
>>>>>>>>> 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
>>>>>>> _______________________________________________
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>>>>>>> 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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>>> xmca@weber.ucsd.edu
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>> _______________________________________________
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> --
> ------------------------------------------------------------------------
> 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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