Math holds a unique position among the natural sciences. It is concrete, practical and, paradoxically, abstract. In all its applications the solution to the problem exists before the problem is presented its practitioners. This effectively dispels any fretting about outcomes, but how to arrive there is another matter.
Simple problems require little effort and due to their simplicity strategies towards a solution are very few. The more complex the problem the more steps are required and many paths towards a solution can be devised. Frustrations are bound to occur at this level. Unfortunately, many get stuck in the realm of the simpler problems and never progress towards the more advanced levels of understanding.
No one is presented a complex problem without first having been through rudimentary training and from there made progressive advancements. The basics will be the foundation for any superstructure they build or discover as they progress towards an answer. Truth be told, it is the teacher who reveals the structures and presents the procedures and how to approach a given problem. The students merely copy the patterns shown to them.
On account of math’s strict logic its laws cannot be broken. Misconceptions and delusions concerning how to proceed in the untangling of a certain problem must be exposed and handled if the endeavors are to yield a satisfactory solution.
When the students undertake the task of solving a given problem they exhibit faith in that their efforts (concerning efforts; it might seem that way to us – perhaps the “agony of faith” is a better phrase) will be rewarded – they believe that a solution exists. It must be noted that it is only those who persist who will arrive at the solution (in math anyway). Stamina and endurance will become increasingly vital as the student conquers higher realms of math.
The student’s confidence in that a solution exists rests solely on their teacher’s word, and the teacher is there to not only teach them, but also correct them and lead them safely towards the desired outcome always encouraging them to believe that the thing is already done – the solution exists.
To the student it might seem as if she or he is in control of and governing the process, but the reality is that it is the commission or task at hand that governs the student and which gingerly, but with a firm hand takes her through its necessary steps always providing new insights into the nature of math.