Whereas the goal of problem solving activities is for students to use some context to better understand mathematics, the goal with [our] applied activities is the exact opposite: to use mathematics to explore how the world around us - the external world that we often think of as the real, real world - works. Instead of discussing which type of activity - procedural, conceptual, or applied - we should use, a more constructive conversation would be about how often and when.  instead of debating which world is the best, we would do better to consider how to best integrate them: how to stop the pendulum from swinging and find its equilibrium (or at least limit the swing to a stabler range). How real-world an activity is first depends on the world in which it exists, and the goal it's intended to serve.