Mathematics—at least formal mathematics—is a type of game. This game is played with symbolic shapes, and uses rules that maintain systematic relationships with a reference class of situations. Generally, formal expressions can be treated as objects that move in constrained ways through discrete states. The form of expressions provides one type of interface through which to engage mathematical meanings. We suggest here that this interface is largely outdated, relying on modes of interaction which have changed little in many decades, and that it is at least plausible that some of the anxiety experienced by many students when they encounter algebra results from the primitive nature of the interface, rather than from the game of algebra itself. We present the results of one experiment conducted in an urban middle school exploring learning in a highly structured mathematics lesson using the Pushing Symbols framework.
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