I think it is safe to say that a synonym for “mathematics” would be “problem solving”. In one form or another you always solve problems whether it is simplifying expressions, solving equations or deriving proofs. When you are solving problems you are supposed to get stuck, otherwise the problem is too simple and it will not challenge you to improve your knowledge or skills. This is why when I meet a new class, my first question will be “what do you do when you are stuck on a problem?” Since I work in junior-high I always get the same answer; “We ask the teacher how to do it”. My follow-up question is of course “if the teacher tells you how to solve the problem, was it really you who solved it and just by hearing the answer, did you learn anyhing or improve any skills?”
The goal in school is to ensure that our students become independent problem solvers and in order for them to become that, they will have to start solving problems on their own. So far all of my students have been used to turn to the teacher for all answers, but now when they will not get answers that easily, they will have to come up with methods that they can start to use when being stuck. They will quickly come up with many ideas on how to make progress, such as discussing and sharing thoughts with friends, looking up information on the internet, drawing pictures, reading the theory in the math book etc. Doing all of this and more, is actively trying to solve a problem.
It is important to guide students on how they can become active participants in their learning, because if students learn how to collaborate and use information from many sources they will quickly develop and improve strategies for problem solving. That is why it is crucial to have a discussion on what prefered activity is and brake it down into a few paragraphs. In this way students know what they are supposed to do and what you expect of them. Furthermore, since activity has been broken down into concrete paragraphs, this will make it very easy to assess students’ activity during class.