Because both reality and abstraction have a prominent role in the teaching - learning cycle in mathematics I have made a  feeble attempt to make objective a couple of notions that seem relatively subjective on first blush.

     In the classroom there are really only two important ingredients to consider when trying to achieve learning, the teacher and the student.

     For the teacher to be effective he needs to spend as much time during each lesson/week/term/year in reality. As a model of this situation I have developed what I refer to as the Reality Quotient. It is a rational expression in which the denominator is the amount of time spent in abstraction, and the numerator, of which, is the time spent in reality. The result is that the more time that a teacher spends in abstraction as compared with the amount of time he spends in reality the less value the reality quotient has. It is my belief that the most effective teachers work diligently to spend as much time as possible in reality. It is my further belief that a great teacher should constantly be trying to keep his Reality Quotient at a value that is at least, one. Mathematically, it looks like this:
R ³ 1,

     One of the results of this effort is that good teachers become great teachers, and once they are great they will be spending at least as much time in reality in their teaching as they spend in abstraction.

      For the student to be effective he needs to spend as much time as possible during each lesson/week/term/year working beyond his ability. As a model of this situation I have developed what I refer to as the Achievement Ratio. It is a rational expression in which the denominator represents the studentís ability and the numerator is the studentís achievement level. The result is that the more time that a student spends working at a level that is above his ability as compared with the amount of time he spends working at a level that is beneath his ability the less value the achievement ratio has. It is my belief that the most effective students work diligently to spend as much time as possible working at a level that is above their ability. It is my further belief that a great student should constantly be trying to keep his Achievement Ratio at a value that is at least equal to one. Mathematically, it looks like this:
     A ³ 1,

     One of the results of this effort is that the student will constantly be working hard enough to have a positive impact on his ability resulting in the need to always work harder still to keep his Achievement Quotient worth at least one. In the end one would hope to have a student who is very bright, and always working at his level of ability.