# [Xmca-l] Imagination or Fantasy

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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

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