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[Xmca-l] Re: Imagination or Fantasy


      Let me try to answer your comments as they occur. I think your concerns are important, but, to a degree, mine are somewhat otherwise.


> On Dec 7, 2015, at  11:34 AM, Huw Lloyd <huw.softdesigns@gmail.com> wrote:
> Hi Ed,
> I imagine that a large part of the problem is the starting place of the
> "mathematical" (in the sense of the contexts you reference) employment of
> space.

I really have no idea as to what problem you are referring to (I can’t even imagine it).

> The relation between the concrete and the imagination can be explored by
> considering imagination as the history of concrete actions and activity.
> This means undertaking measurements (i.e. mathematics with units) in the
> contexts of problems to achieve 'concrete' notions of quantity etc
> (Davydov).  Imagination can then be used to access and orient to these
> historically achieved understandings.

Your definition of imagination as a sort of history does not work particularly well for me nor has too much support in the literature. I have thought much about Davydov and although I believe the reports and papers I have pursued, I am not particular convinced that using notions of quantity necessary is a panacea. I see similar results in other classrooms without that particular curricular focus. That said, there are characteristics of Davydov curriculum which are, one might say, necessary (it is a bit hard to tell as it is hard to find accounts of .extended classroom interaction or even a teacher’s text). However, more to the point, I would not necessarily use notions of quantity with, for example, sixth graders who lacked the benefit of a Davydov curricular approach in the first grade. I would, however, use something ‘stable’ and I would focus on mathematical structure.
> The implication here is that the space of imagination is the space of
> relations, and that the abstract 'mathematical space' is (initially at
> least) an obstruction.  Note that problem (activity) oriented actions are
> important to facilitate the synthesis of a richly structured space (set of
> relations).

I admit to not knowing where you got the idea of an ‘abstract mathematical space.’ I know that people sometimes say things like this, but it has always seemed rhetorical. I also know that people like to talk about a richly structured or connected space of relations. I don’t have a problem with this, but the grain size seems too large for the questions I am asking.

> Presumably, the teacher in compliance with the package of education in
> busily propagating the formal patterns of 'maths' which has little to do
> with the process of doing maths.   (Problems are not usually part of the
> package of education).

I have no idea why you brought this up although this is true of some teachers (often I might add since they don’t know otherwise so perhaps it is less compliance than you think)

> According to this manner of reasoning, the reason why some of the kids
> understand the lessons is that they are either well-practiced in the
> "concrete" manipulation of objects or well drilled in formal pattern
> matching.  In other words, their capacity to do maths is more a function of
> their home environment than what they encounter in school.

There are lessons and there are lessons. If the purpose of a lesson is to reproduce the thinking of the teacher/text, then what you say makes sense. This goes for Davydov’s curriculum as well as others. 

> I think you'll agree that from this vantage that space, imagination and
> orientation are all cognate terms.  The problem (which is the same problem
> as the zoped) is that the packaging of education "invades" this space,
> rather than helping to construct it.

I seem to be in another conversation, this time about the packing of education. I am okay with you taking this vantage, but I don’t think it is that simple so I tend not to do so. Perhaps a story is in order. A number of years ago I heard a fairly well known early childhood educator speak. She had gone to Berkeley for her teachers training and picked up the vantage you seem to speak of (and there is nothing wrong with that). She went into teaching - and loved it - and made worthwhile efforts to act out her new convictions. About three-quarters of the way through the years, a colleague and experienced teacher asked her if her class knew, say 'thingy x’ as the next year it would be assumed they did. She realized that they didn’t as her focus had been otherwise and there was no time to do something about it. She took what she view her failure quite seriously - and I happen to agree as I don’t think you can do this to kids no matter what your good intentions - and went back to graduate school to pick up what she had missed. Maxine Greene relates a similar situation in “Oneself as a Stranger.”

My point is not that you are wrong, but I am a teacher and, while I agree without reservation the ‘packing’ is more than a problem, I choose to support those who are doing something about the packing’ while focusing my attention on the child who sits in the classroom we two share.

>> From what I can gather, packaging is an idea of capitalism and that the
> 'truer' an education is, the freer it is from packaging.  Nevertheless,
> without the problems of packaging one would not be in a position to
> appreciate an idea in distinction to the habits, dogmas and packages of
> one's times…

‘Packing' isn’t the problem I find myself most concerned about (and I am not arguing it is less important); teaching in the now is. This is a moral decision on my part.

> Best,
> Huw
> On 4 December 2015 at 19:03, Ed Wall <ewall@umich.edu> wrote:
>> All
>>     For various reasons I have been thinking about a kind of imagination
>> that might be subsumed under statements like “assume that,” “let,” or
>> “Imagine that” (and these may be, in fact, very different statements
>> although, under certain circumstances, might be the same.” In doing so I
>> came across something written by Vygotsky in Imagination and Creativity in
>> the Adolescent (ed Rieber) p163: “It is characteristic for imagination that
>> it does not stop at this path, that for it, the abstract is only an
>> intermediate link, only a stage on the path of development, only a pass in
>> the process of its movement to the concrete. From our point of view,
>> imagination is a transforming, creative activity directed from a given
>> concrete toward a new concrete.”
>>    I find this quote very interesting in view of a previous discussion on
>> the list regarding Davydov’s mathematics curriculum in that I am wondering
>> whether part of what is going on is that children are being asked to
>> “imagine." I have other mathematical examples of this join the elementary
>> school that are possibly a little more obvious (if somebody is interested I
>> can give them off list). Anyway, one reason for my wondering is that for so
>> many people mathematics is not concrete; i.e. there is no stepping from
>> concrete to concrete; the sort of get stuck, so to speak, in the abstract.
>> So let me give two examples of what I am wondering about and then a
>> question.
>>   My first example:  It is possible that we would all agree that to see a
>> winged horse is imagine a winged horse as there is no such thing. In a
>> somewhat like manner, a simple proof that the square root of two is not a
>> fraction begins with “Assume that the square root of two is a fraction.”
>> This is not so thus, in sense, one must imagine that it is true and then
>> look at the consequences (the square root of -1 is perhaps another
>> example). This seems to be a case of concrete to concrete through
>> imagination and this type of proof (a proof through contradiction) seems to
>> be very hard for people to do.
>>   My second example: The teacher goes up to the blackboard and draws
>> something rather circular and says “This is a circle.” She then draws a
>> point somewhat towards the center of the planar object and says, "This is
>> its center.” She then says “Every point on this circle (waving her hand at
>> the object on the blackboard) is equidistant from the center.” None of this
>> is true, but somehow we are meant to behave as if it were. Each step here
>> seems to go through imagination from the concrete to the concrete. (Hmm , I
>> see that I am really saying from the physical concrete to the mathematical
>> concrete. Perhaps Vygotsky wouldn’t allow this?)
>> [I note by the way Poul Anderson took on the consequences of a winged
>> horse].
>>    So my question, as Vygotsky seems to identify imagination with fantasy
>> (this may be a fault of the translation), what would Vygotsky have called
>> my examples? A case of sheer conceivability or something else? There is, I
>> note, good reason to call it imagination, but I’m interested in your take
>> on what Vygotsky’s take might be.
>> Ed Wall