Play and mathematics

When I first read this passage, I wasn’t sure what Vygotsky meant.

In the view of earlier writers, the imaginary situation was not the criterial attribute of play in general, but only an attribute of a given group of play activities. I find three main flaws with this argument. First, there is the danger of an intellectualistic approach to play. If play is to be understood as symbolic, there is the danger that it may turn into a kind of activity akin to algebra in action; it may be transformed into a system of signs generalizing actual reality. Here we find nothing specific in play, and look upon the child as an unsuccessful algebraist who cannot yet write the symbols on paper, but depicts them in action.

The word “symbolic” is what began the point of my confusion, but I continued reading. Several pages later, I understood what Vygotsky meant.

Nevertheless, properties of thing as such do have some meaning: any stick can be a horse, but, for example, a postcard can never be a horse for a child. Goethe’s contention that in play any thing can be anything for a child is incorrect. Of course, for adults who can make conscious use of symbols, a postcard can be a horse…. For a child [a match] cannot be a horse: one must use a stick. Therefore, this is play, not symbolism.

This reminded me of our discussion last week but also sparked my thinking about mathematics: is mathematics (algebra in particular) a version of play? As adults, we can use a very abstract symbol (such as x) to say “Let x be ____.” I can imagine letting this symbol be any noun within my vocabulary and can imagine imbuing x with all of the properties and characteristics known to that noun. But the critical point here is that we need to be at the level of development to make conscious use of symbols for “any thing to be anything."

This also reminded me of a practice that we do in math where we use certain manipulatives (such as counters, blocks, sticks) to represent the things being discussed in a problem. Only very rarely in math will we actually use the very physical objects that are being discussed. For instance, “Suppose Tom has 54 watermelons….”

There is a critical point, according to Vygotsky around age 3, at which symbols (but not just any symbols – symbols with some resemblance to the actual objects) can be used as indicators of the real thing. My guess would be that a drawing of a watermelon would be a suitable symbol for most children, but at some point (later in adolescence) just a circle could be used, or a line, or a dot, or x.

This reminds me of “period 4” from our discussion last week, the point at which symbols can be dissociated from their original context, internalized, and used as a tool in problem solving.

Back to the question I asked above: is mathematics play? Vygotsky says play is “such that the explanation for it must always be that it is the imaginary, illusory realization of unrealizable desires…. Play is essentially wish fulfillment – not, however, isolated wishes, but generalized affects.” With this latter characteristic of play, I would have to say no, mathematics is not purely playful, as I would not characterize mathematics as being wish fulfillment. It is to a large degree imaginary, though.

Vygotsky’s argument that [Games with imaginary situations] ó [Games with rules] reminded me again of our discussion of experiences. All imaginative and creative acts are combinatorial products built from our experiences. A child pretending to be a mother must assume the role of mother in a way that she is familiar with – a way in which the child has experienced (or imagined) a mother figure being. The child's experiences act as constraints on the kinds of rules that may be placed on their acts of play. The conclusion from this seems to be that the greater quantity and quality of experiences a child has, the more creative it can be about the "rules" it ascribes to its imaginary and playful roles. Of course, once the child reaches period 4, "any thing can be anything" (but still within the confines of one's experiences).