[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]

[Xmca-l] Re: Maths and science in Korea


      Interesting what you say here about 'combinatorial imagination' and number. There is a mathematical sense, in which even relatively  small numbers scan be treated as grammar, for instance, sixty. That is, besides its notation in base-ten, it has importantly the factors of 1, 2, 3, 4, 5, 6, 10, 12,  15, 20, 30. 60. I can see that one might think that within physics the combinatorial aspect of number might become a fetter (although the primes seem to have a sort of importance). However, the combinatorial properties of number are of more than a little important within modern technology and, of course, modern mathematics (and even calculators - smile). Anyway, I agree there is a sense in which larger numbers seem more complex (an interesting question might be whether 60 is more complex than 100000000000000001).
      I appreciate your age example but there is a problem. The months do not name the same number of days and hence, I would hope that neither a child or an adult would convert to 'fractional' months to give an 'exact' answer. However, if one was to convert to weeks, fractions would make sense. I don't think many adults I know would convert to fractions if I asked them to be exact (and I don't think any hesitation would be due to difficulties with making fractions from months). What I tend to do (and I suspect you will find this peculiar) is to subtract the year I was born from the current year (I remember those - smile) then treating a year as an interval divided roughly in fourths, give a rough fraction (saying about). This, by the way, is roughly what younger students do although they count up rather than subtract (and there is much discussion as to how one counts the year of birth) and use "just", "one-half", and "almost." Fractions are, more or less, a consequent of the multiplicative properties of numbers and, in a way, they still may be within the sphere of 'combinatorial imagination'; e.g. the importance of the factors of 60 in early arithmetics. However, I am more than sure I imperfectly understand this notion so this is just a wondering.
      Yes, I left off the Sino- as I was interested in local variations and I agree that eleven and twelve is a matter of lexis although all the numbers between 9 and  20 (and they are more than critical for the operations of arithmetic) need, in a sense, to be lexicalized. The problem seems to be students need to learn these structural exceptions and that takes, one might say, precious time. There is also the possibility that the structure of the base-ten number system is eventually weakly grasped and that what remains tentative is passed over in the rush of instruction. I do know there is a lot of evidence that elementary school teachers in the US have a weak grasp of the structure of the base-ten number system. Oh, they can count and do all the operations fine, but are actually fairly unsure what it is all about (and, of course, this may be the case with Korean elementary school teachers also). 
     Reading your second example reminds me that I don't quite understanding how you are using combinatorial (I may be mathematizing it too much). I agree that people tend to misrepresent large powers of ten (and most have even greater problems with 'large negative' powers of ten). As I have a reasonable background in the physical science and biological science, I have tended to interpret this as, of the most part, a lack of context or, perhaps, experience (However, I may be quite wrong). Anyway, I see a thousand as 10^3, a million as 10^3 x 10^3 and so on so the commas actually mark the naming. Thus I see the naming as cultural, but corresponding to mathematical notation.
     Anyway, I agree with your point about 'making strange the way they do it.'  For example, I remember a third grade classroom where a young boy remarked, in essence, that our characterization of the numbers 6, 10, 12, 14, etc. in terms of even was strange. However, his reasons were combinatorial. Perhaps, you are saying that merely having the ability to perform or notice the combinatorial is insufficient guarantee that this move of 'strangeness'  can be performed. I find this quite interesting and I would definitely agree (I remember when I first consciously realized this). I have struggled for years to inculcate such tendencies in my students. However, I've so far been unable to sufficiently frame what I seem to be chasing.


On Dec 22, 2014, at  3:31 PM, David Kellogg wrote:

> I think in any language, big numbers are just an example of what Ribot was
> calling "combinatorial imagination" (see earlier thread on imagination).
> That is, all languages treat the most commonly used numbers as lexis and
> the larger numbers as grammar--so for example in English the numbers one
> through twelve are all "molar" in the sense that they are single morphemes,
> while Avagadro's number (six hundred and two hextillion, two hundred and
> fourteen quintillion, one hundred and fifty quandrillion) is decomposable
> into base ten units at every point. This is part of a the much more general
> property of language which means that the words which realize scientific
> concepts (e.g. "immersion") are morphologically complex while the ones
> which realize everyday concepts (e.g. "dunk") are morphologically simple.
> One of the problems with Chomsky's model of language (the one to which Roy
> Harris is pointing, actually) is that it places this kind of combinatorial
> imagination at the centre of the language system and considers fixed idioms
> to be epiphenomenal, while in everyday life, that is, in the vast majority
> of language uses, it's really the other way around: we use language as a
> set of fixed (but refixable) expressions and reserve our creativity for
> other problems.
> Engeström's book "Learning as Expanding" begins by attacking Gagne on
> combinatorial imagination; Engeström argues that all learning is
> essentially expansive and not combinatorial. I have always thought this a
> mistake: as Vygotsky says, development always means the introduction of
> something new, something not present even embryonically at the outset, and
> it's for this reason that a historical account of a process can only be
> teleological and cannot actually be predictive (we can't predict learning
> any more than we can predict evolution). So I can easily imagine that
> combinatorial imagination is a big step forward at one point in the
> development of the number system and then it becomes a fetter on the
> imagination because it focuses attention on how the number is composed
> rather than on what we can do with it (we actually don't use the long form
> of Avagadro's number when we talk about  chemistry; we just say "Avagadro's
> number" or use a calculator).
> Here are two examples of number problems that I think require more than
> combinatorial imagination. The first is the one that I clumsily confused
> you with in my last post. Suppose a child is nine years and four months
> old, and I ask the child how old she is. She says "nine and...."  And what?
> The adult way is to say nine and four months, but that doesn't tell us how
> many years old the child is, that is, how many fractions of a year lie
> between the child and the child's next or last birthday. In order to answer
> that question, the child has to convert months to twelfths of a year, and
> then try to convert these to thirds or decimals. This requires more than
> combinatorial imagination; it involves reframing the problem as one of the
> imperfection of the non-decimal system we use to calculate months.
> The second stems indirectly from the problem you raise. Although I think
> the difference between the Korean (actually, Sino-Korean) numbers between
> ten and twenty is negligible (it's just a matter of lexicalizing two
> numbers, eleven and twelve), there is a very serious problem that dogs even
> advanced learners of English here. The Korean won is a very small currency
> unit--there are about a thousand to the dollar at the moment. So to express
> the price of anything worth buying, you need to talk, in English, in units
> of ten to the third power (a cup of coffee costs thousands, salaries and
> rents are in millions, cars costs tens of millions, and real estate costs
> billions and trillions). But the Korean number system uses units of ten to
> the fourth bower ("man" means ten thousand, "ok" means a hundred million,
> and "cho" is a billion). So when you read statistics in the English press
> they are very often wrong by a factor of ten. This requires more than
> combinatorial imagination, because it involves realizing that the placement
> of commas in a large number is just a cultural convention corresponding to
> language and not to mathematics itself, and also involves "making strange"
> the way we do it, and thinking instead in terms of the way they do it.
> On Sunday our Vygotsky seminar met at my apartment to proof the galleys of
> our new book of Vygotsky's lectures on pedology, and during the lunch break
> some of my former students were looking at the paintings I did twenty years
> ago. At the time I was obsessed with "point of view", and all my paintings
> featured some intrusion of the artist's own body--usually an arm or a pair
> of spectacles. I guess the purpose of it was to "make strange" the act of
> looking at a painting and to instead force the viewer into my own
> viewpoint. But it also had the curious property of making strange the act
> of painting a painting, because I could never get over the fact that both
> the part of my own body and the rest of the painting were in equally sharp
> focus. That's not the way point of view really works!
> David Kellogg
> Hankuk University of Foreign Studies
> Korean
> On 22 December 2014 at 08:29, Ed Wall <ewall@umich.edu> wrote:
>> David
>>      This is quite interesting. Let me give some different takes on some
>> of what you write (I, by the way, am still unsure)
>>   Some say part of the fetters you refer to are in the manner in which
>> the equal sign is used (e.g. as compute) and the propensity for vertical
>> addition in the early grades. I have seen a US 3rd grade teacher complicate
>> things somewhat (she and the children were definitely working with
>> countable objects) in starting with an unknown quantity, remove a known
>> quantity, and, by obtaining a known quantity, figure out what the initial
>> whole was without too much trouble. That isn't too say that things might
>> not have proceeded more efficiently with another curricular starting point.
>>     There is an interesting different between Korean students and US
>> students in the early grades. US students have something called eleven and
>> twelve and Korean students have something translated, in effect, as ten-one
>> and ten-two. There has been speculation by some that these can be fetters
>> of a sort on the way to decimal number (there are also things called nickel
>> and quarter and five dollars not to mention inches, etc. which can also be
>> somewhat un-helpful).
>>     On a different note, what is the adult system of determining how old
>> one is in years, exactly? I know how young children do this correctly, but
>> inexactly.
>> Ed
>> On Dec 21, 2014, at  4:50 PM, David Kellogg wrote:
>>> Let me float a hypothesis, and see what Huw and Ulvi make of it. A
>> learning
>>> activity (any learning activity) is best described not as a synoptic
>>> hierarchy of molar units like operation, action and activity. Viewed
>>> diachronically, from the point of view of psychology, a learning activity
>>> is a non-hierarchical historical sequence, such that any given
>>> "method" eventually, in time, turns out to fetter progress and must be
>>> discarded, and the end result is not an inter-mental social form of
>>> activity but instead an intramental psychological one.
>>> Take the Schmittau work that Ulvi references as a concrete example.
>>> Schmittau showed that the American curriculum (like the Korean one)
>>> introduces the notion of number by counting separate objects. This allows
>>> the child to grasp the number very concretely and quickly. Groupings are
>>> then introduced, and this corresponds once again to what we see children
>> do
>>> naturally (see Chapter Eight of HDHMF). So at every point the American
>>> curriculum takes the line of least resistance. But that means that at a
>>> specific point, the notion of number based on concrete, separable objects
>>> becomes a fetter on the child's progress. Schmittau locates this point
>>> quite precisely: it's the moment when the child, accustomed to add known
>>> quantitites of physical objects together to obtain an unknown quantity,
>> is
>>> asked to start with an unknown quantity, remove a known quantity, and, by
>>> obtaining a known quantity, figure out what the initial whole was (e.g.
>>> "I made a bunch of snowballs and put them in the freezer. I threw one at
>> my
>>> big brother at a Christmas pary, and two at my friends when they teased
>> me
>>> at New Years. Now I have only half a dozen left for April Fools Day. How
>>> many snowballs did I make?")
>>> Chapter Eight of HDHMF asks the question of whether "arithmetical
>> figures"
>>> (that is, physical groupings of countable objects) will keep the child
>> back
>>> from learning the symbolic manipulations afforded by the decimal system
>> of
>>> writing digits, or whether they will naturally evolve into the decimal
>>> system (because the children will of their own will invent a physical
>>> grouping of ten objects). Interestingly, Vygotsky concludes that any
>>> experiment along these lines would be unethical (and THERE is a
>>> correspondence with Chomsky, who has often correctly noted how one of the
>>> things that keeps linguistics in a "paper and pencil" era corresponding
>> to
>>> sixteenth century physics is the immorality of experimentation on human
>>> subjects). But, like Chomsky, he resolves the question with paper and
>>> pencil (in Chapter Thirteen) with a very amusing MIS-reading of
>> Thorndike's
>>> "Psychology of Arithmetic".
>>> Thorndike is criticizing the way in which our parents and grandparents
>> were
>>> taught arithmetic as a symbolic system akin to language. Vygotsky
>>> apparently doesn't get Thorndike's irony, and thinks that Thorndike is
>>> lauding this culturally approved method over Lay's newfangled system
>> based
>>> on "arithmetical figures" (dominos, in fact). See the attachment: it
>>> involves analyzing a picture where there is one girl on a swing and
>> another
>>> on the ground ("How many girls are there?") a kitten on a stump and
>> another
>>> on the ground (which Vygotsky misremembers as dogs). And so, by a process
>>> of misreading and misremembering, Vygotsky turns Thorndike into a
>>> cognitivist. Thorndike would probably rather be a dog.
>>> Interestingly, the way Vygotsky resolves the whole dispute is
>> similar--that
>>> is, the child triumphs not through the adequacy of his or her own method
>> or
>>> through seeing the superiority of the adult method, but rather through
>> the
>>> inadequacies of both. For example (and this is my example), a child with
>> a
>>> notion of number based entirely on separable objects has a very hard time
>>> measuring how old he is in precise terms. On the other hand, the adult
>>> method of measuring years out in months is NOT a decimal method. The
>> child
>>> therefore has to grasp and perfect the adult system just in order to
>> answer
>>> the simple question--how old are you in years EXACTLY?
>>> David Kellogg
>>> Hankuk University of Foreign Studies
>>> On 22 December 2014 at 07:01, Huw Lloyd <huw.softdesigns@gmail.com>
>> wrote:
>>>> Ulvi,
>>>> The essential 'method' is to facilitate students'  own experimentation
>> with
>>>> methods.  This is called learning activity.
>>>> Huw
>>>> On 21 December 2014 at 12:15, Ulvi İçil <ulvi.icil@gmail.com> wrote:
>>>>> Hello,
>>>>> I know there are some works comparing Russia (Davydov's curriculum) and
>>>> US,
>>>>> and even some works done in US with an application of Davydov's, e.g.
>> by
>>>>> Schmittau.
>>>>> I would like to know, not in detail, but just in general, which main
>>>>> factors lie behind this success in Russia, it is Davydov, or Zarkov or
>>>> any
>>>>> other scholar's method.
>>>>> Thanks in advance,
>>>>> Ulvi
>>> <For Ulvi and Huw.docx>