Just as I clearly formed that question in my brain, I came across Jesper Juul's definition of a game in Karen Collins' Game Sound: An Introduction to the History, Theory, and Practice of Video Game Music and Sound Design. His definition includes:
- Variable & quantifiable outcomes
- Player exerts effort to influence outcome
- Player feels emotionally attached to the outcome
I think the first three parts could describe an activity or a game, but it is the last one that distinguishes between the two. Engaging students mentally is one thing, we should all try to do that. But sometimes I think we will want to engage them emotionally, to make them feel a strong desire to do well. And that makes a game.