Below is a link to studies from the University of Kansas. The extracts here are not about rewards but the use of sanctions and cover 'related, respectful and reasonable consequences' for student behaviour. The study goes on to gives an interesting account of logical consequences v punishment. It also has lots of good examples..

"What are logical consequences?

Logical consequences do not naturally occur as a result of behavior, but are intentionally planned by teachers and administrators. Logical consequences are similar to what would happen to an adult in a similar situation, therefore teaching students skill that they will need to be successful later on in life. Logical consequences need to be related, respectful, and reasonable (Nelson, 1985).

  1. Related

    Related means that the consequence is clearly connected to the student’s behavior and its function. This requires the teacher getting to know his or her students well and becoming proficient at the process of both formal and informal functional behavioral assessment and analysis which is discussed in detail in another section of Special Connections. The function of the same behavior may be different for each child, or even for the same child at different times. Therefore, it is crucial that teachers take this into consideration and do not use one consequence, such as isolating the student from others, for all inappropriate behavior. An example of a consequence that is not related would be having a student go to time out for calling another student a name. A related consequence may be to have the student spend some of their free time (recess or after school) discussing the natural consequences of the action with the teacher (e.g. hurting that person’s feeling, possibly getting called a name back) and writing the offended student an apology.

  2. Respectful

    Consequences need to be given with empathy in a respectful voice tone. If not, the student will focus more on the feelings of the adult and perhaps their own feelings of anger resulting from being talked to disrespectfully and not reflecting on their choices. The student that is not treated respectfully often becomes aggressive, passive, resentful, and/or uncooperative and may try to get revenge against the teacher.

  3. Reasonable

    Reasonable refers to not providing consequences for a student’s inappropriate behavior that is too severe. It is not reasonable to require that a student lose all their recesses for the week for being silly in class or to lose the next month’s field trip for getting in a fight on the playground. A more reasonable consequence may be to stay in for one recess to practice appropriate classroom behavior

Is the philosophy of positive behavioral supports and interventions against the use of consequences?...