In this case, the reasoners are the operators which are available and the beliefs concern the equation to be solved and general knowledge, for example, about arithmetic.
Monitors are concerned with the application of reasoners.
Once such an outline is sketched it becomes a challenge for greater precision, which is the hallmark of the approach of computational mathetics. For example, we might immediately ask what is meant by abandon-path , and begin to elaborate ideas how about an agent changes the equation currently focussed upon. At the moment, we have an informal picture of an agent being ascribed monitors to account for behaviour such as deciding which operator to apply, setting up a plan of action, abandoning a path, and so on, the monitors themselves being represented in a production system-like notation. We can make further general points, however. For example, which items should be ascribed to which level is not fixed but depends upon the agent.