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[Xmca-l] Re: Imagination or Fantasy

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

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.

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

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).

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.

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.

>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...


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