As elementary students begin to program using tangible blocks, they must coordinate their use of counting with the movements, directions, and numbers they use to move a character. In our study, we analyzed 13 first graders' first attempts at coordinating these elements when playing a programming game on the iPad that used tangible programming…

The multiplication principle is a fundamental principle in enumerative combinatorics. It underpins many of the counting formulas students learn, and it provides much-needed justification for why counting works as it does. However, given its importance, the way in which it is presented in textbooks is surprisingly varied. In this paper, we document…