How would I use logical positivism to explore why teachers teach (objectives), what should be taught (curriculum) and how should the curriculum be taught (teaching strategies)?
The scope of a positivistic inquiry is limited to what is susceptible to observation. That observation typically is in terms of relations or interactions between the elements involved. Accordingly, the inquiry cannot ascertain notions such as a teacher's ulterior motive for teaching, nor the contents and structure of an objectively ideal curriculum. That is because such notions are inextricable from intuition, and from the impossible grasping of the inner cause(s) or substance of something external to the observer. This leads to answers which are rather of a pragmatic and/or utilitarian nature.
By way of example, the answer to "why teachers teach" may point to the foreseeable socio-economic benefits (i.e., prestige, salary) that tend to improve teachers' position or circumstances. That tentative answer will be either reinforced or weakened as [in-]consistent with subsequent observations.
Positivism is not infallible, though. A teacher's ulterior motive might be something else (for instance, an aversion to any other professions). The teacher's ability to conceal from everyone that ulterior motive would mislead the inquirer's interpretations of the relations observed. But Positivism is straight-forward by explicitly excluding beforehand the aspiration of ascertaining the essence or substance of the matter at issue.