Abstract:

Using grading scales has a long tradition in higher education and on the job training. It is used in a variety of ways to assess what students / trainees have learned and how well they can apply their newly acquired knowledge to accomplish certain tasks or solve given problems. The assessment result may take the form of a written report by the assessor indicating how many tasks / problems the assessees has successfully accomplished and why they failed on the other ones. Students or trainees can use that information to prepare for a rehearsal of the same test or assignment, or for the next one. In the case of a final examination, assessors may use it to justify and report their decisions regarding the status of the assessees at the institution or workplace.

Using assessment protocols, including ordered grades on a bounded grading scale, can be both challenging and inefficient to work with, when it comes to aggregating the grades of a portfolio or sequence of assessments. Also, assessors find it often difficult to decide which of two successive grades better represents a given assessment result, when the grading scale is rather coarse, e.g., A up to G, or 1 to 10. This may explain the invention of alphanumeric grading scales with special codes like B− and C+ to solve the grading conflict.

It is clear that such workarounds can’t go on forever. Here, the generalized concept of a bound-ed grading scale between lower bound LB and upper bound UB is proposed as a solution. In other words: a grading scale no longer consists of distinct alphanumeric codes; instead, a grade may be any point on a line segment from LB to UB. By associating numbers with LB and UB, the grading scale can be simply denoted by [LB,UB]: the ordered range of real numbers between LB and UB.

In this paper, we will show how to calculate with grades on a bounded scale [LB,UB] such that the sum of two grades is again a grade, and any grade may be multiplied by a positive decimal number, called its weight, to obtain a weighted grade. We will then show how sums of weighted grades play the same role on bounded scales as the more familiar concept of the weighted average of a list of numbers. Furthermore, we will introduce the concept of a neutral score, which marks the threshold between “negative” and “positive” grades. By default, the neutral score equals the midpoint between LB and UB, but we will show that any grade may play this role, granted a simple adaptation of the weighted sum of grades.

Bounded grading scales together with the above-mentioned operations may be used to solve several problems that emerge quite commonly in educational assessment.

We will demonstrate this with three scenarios:
(1) A complex assignment shall be assessed using many criteria all using the same bounded scale. Then the overall assessment may be reported as the sum of weighted grades over all criteria.
(2) Same as before, but now each criterion is associated with its own bounded scale. Then, the solution will be to map the grades to the percentage scale and to calculate a weighted sum of the percentage grades.
(3) In Group-Peer Assessment, the assessor calculates the grade for a group product (using 1 or 2), as well as the individual student contributions on the signed percentage scale based on mutual peer ratings. Then, a rather simple scoring rule suffices to map student contributions to student grades on a constrained percentage scale such that the mean student grade equals the group grade.

Keywords:

Bounded scale, grading scale, sum of grades, weighted grades, weighted average grade, scale mappings, percentage scale, signed percentage scale, constrained percentage scale.