Modern Teaching Theory 101

It is common knowledge that in some schools, teachers' performance bonuses are tied to the students' Mean Standard Grade (MSG) , the grade point average of the students. If the average is higher than par, then your pb will be higher. We shall explore the prospect of a rational way to optimize return given fixed effort, where effort is defined as the number of overtime hours that a teacher puts in to help the student outside of normal curriculum time and return is defined as the percentage improvement of the MSG (with respect to the previous MSG). Subsequent entries shall explore the possibility of the existence of a convex efficient frontier (since return-effort changes in a non-linear fashion as individual weightings are varied) where effort can be optimized (read: slack) given fixed return (say, just to maintain a C performance grade). (ok, i just typed that to impress. Who's impressed?).
We can express an entire class of 40 students' grades at any point in time as a vector with 40 entries, (x1, x2, x3, ..., x40), with the position of each entry representing the grade of a particular student. The average of these 40 variables will be the MSG. Firstly, we assume that an unit of effort expended on, say, student A will produce a percentage change in grades which is almost always different from a corresponding unit of effort expended on student B. In other words, the same amount of time spent on tutoring different students will produce different results. Secondly, we assume that it is almost always harder to increase an A2 student's grade than it is to increase an F9 student's grade. In other words, it is harder to help someone improve from 70 points to 80 points, than it is to help someone improve from 10 points to 20 points. Subsequently, *cue drumroll* it is intuitive that each unit of effort should be expended on students who can produce the greatest marginal increase in grades. (For the weak reader, please refer to proof * below). So it is plausible that the majority of teachers' approaches are not producing the maximum returns (unless you're a pessimist who thinks that teachers expend zero units of effort in the first place, which is already an efficient position).
This conclusion is rather startling, in the fact that teachers tend to spend a lot of effort on the 'good', attentive students who have demonstrated fantastic attitude (relative to the rest of course) in class with consistently good grades, but possesses considerably less 'up-side', less scope for improvement in their grades. We propose a change in the strategy of teachers in that more effort should be directed at the weakest students, as they will bring about a greater increase in the MSG.
Of course, all these rest on our second assumption, that it is easier to bring F9 up, then to bring A2 up. In our above expose, we have tagged the effort required to the grade of the student, that a F9 student requires a certain amount of effort, and a A2 student requires another certain amount of effort, but in reality, this amount of effort often depends on the attitude and motivation of the student. It could very well be the other case, where an F9 student is unmotivated and a 'blackhole' - no amount of effort spent is going to produce results. Hence, a more detailed analysis would include the students' grade, level of motivation, probability of getting motivated, prior history of improvement (momentum) etc. However, given the optimistic nature of teachers in general, coupled with the general sentiment that 'there is no such thing as students who cant learn, only teachers who cant teach', i think the second assumption is reasonable. There are certain extreme cases in which the proposed strategy is obvious. For example, in a class of 39 F9 students and one A2 student, it is quite clear that a lot more effort should be spent identifying that one F9 student (or two) who has the most potential and helping him/her than to help that A2 student.

(* Proof: For arguments' sake, suppose that we started with 38 students with 50/100 points for their first test, student A with 10 points and student B with 80 points. Hence the MSG is 49.75. Given our second assumption (and for simplicity), we assume that 1 unit of effort is required to boost student A's by 10 points, and 2 units needed to boost student B's by 10 points (In theory, we should posit a hyperbola or exponential for the effort required to boost a students' grades to reflect the increasing difficulty, as the grades get higher, in improving the students' grades). Suppose we adopt a strategy of increasing student A's and student B's grades by 10 points each, (A: +100%, B: +12.5%), we would have increased the MSG to 50.25, an increase of 1%, with an outlay of 3 units of effort. However, if all three units of effort is spent on A, we can boost student A's grades to 30 points, boosting the MSG to 50.5.)