Grading &

The Dilemma

     At the end of every grading period, the refrain is the same: "But a 79.3 is my average. Can't you give me an 80?" The meaning here is clear. With just a little changing here or there, the 79.3 could be raised to a 79.5, then the "rounding rule" takes over and the 80 is the result.

     The scene is repeated many times all over the school. The reason? The 79 is assigned a letter grade of C while the 80 is assigned a B. As we all know, a B-student is "better" than a C-student. But what is more than that is how it affects a student's GPA---the famous (or infamous) Grade Point Average. A grade of B earns 3 points toward one's GPA, while a C earns only 2 points. The traditional (integer) scale looks like this:

A 4 points
B 3 points
C 2 points
D 1 point
F 0 points

     Now, in the world of sports, one point can make the difference being world champion, or just a forgotten "also-ran" who finished in 2nd place. But it is unfortunate that we educators must play the most important "game" in life---teaching and imparting knowledge---by the numbers. After all, is a student who gets a 79 on his report card really so very different from another who receives an 80? Any teacher knows that such is not the case. However, the latter grade has the decided effect of doing the reverse: B is better than C. Of course, a true B-student should be considered as more capable than a true C-student when measured over a long period of time. But students tend to look at the here-and-now of their individual situation, for natural reasons, rather than look at the long trend. As educators, we tend to see the big picture on two planes: time and large student populations.

The Purpose of the GPA

     But let us return to a more fundamental question: Why do we have GPA's in the first place? Basically it is for administrative purposes, to govern grade advancement and graduation. It is a measuring device, a ruler, to see if the student "measures up" to certain standards. If a student has taken a given number of course credits and in the process has earned a certain minimum GPA, he is allowed to remain in the institution, advancing to the next grade level, or graduate. If not, he is retained (usually on probation) until the minimum GPA is attained or else he is asked to leave that particular institution and seek his education elsewhere.

     Is this fair? Is this education? Those are certainly $64 questions. Many have argued convincingly on both sides of the issue and the debate rages on with no easy solution in sight. So I doubt this paper can resolve it either. What this paper does purport to present is an alternative to the unfair categorization of students into the A, B, C, D, and F slots that we have placed them for so long, and especially as it relates to GPA's.

Illustrative Case Histories

     Let's take a typical John Smith sort of student and see what happens to him at the end of the year when his final grades come out. His report card might look like this:

COURSE % Grade G.P.
English 86 B 3
Math 84 B 3
U.S. History 73 C 2
Chemistry 69 D 1
Psychology 91 A 4

     He has a total of 13 grade points (sometimes known as quality points) and a GPA of 2.6. This translates into a C+ quality of work, using the scale above. However, if we take his percent grades and perform the same sort of arithmetic, we obtain a value of 80.6, which is equivalent to a B-, virtually a 3.0 GPA quality. This doesn't seem fair to John, does it?

     As I understand it, our school has taken a step to alleviate this sort of statistical imbalance by making a finer gradation on the translation of percent grades to quality points. The scale is as follows:

Range Letter Points
100-90 A 4.0
89-85 B+ 3.5
84-80 B 3.0
79-75 C+ 2.5
74-70 C 2.0
69-65 D+ 1.5
64-60 D 1.0
59-0 F 0

     This system is obviously more just to a student when certain grades (B's, C's, or D's only) fall into the top half of a given letter category. If we apply it to John's grades, he gains one additional quality point (a half-point each from English and Chemistry). His new total of 14 points now computes to only a 2.8 GPA. Higher, yes, but still below the percent value obtainable by using the raw scores.

     Now, let's examine the case of Mary Jones, a classmate of John's. Here are her grades:

English 85 B 3
Math 81 B 3
U.S. History 74 C 2
Chemistry 65 D 1
Psychology 90 A 4

Under the older integer system, she has the same GPA as John: 2.6. But her percent average is only 79.0, a figure equivalent to 2.9 quality points (see Chart I below).

     Two things are notable here. First, although the GPA figures were identical, she does seem to be slightly weaker than John, based on percent scores. And any teacher knows how difficult it can be at times to raise one's average one or 2 points in the course of a grading period if there are frequent tests and quizzes given. The divisor being larger requires a greater, more sustained effort on subsequent tests than most students and parents---yes, and even administrators---sometimes realize. So John deserves some recognition for his 80.6 over Mary's 79.0, eventhough both students are "doing well", making normal progress. In fact, maybe on one or two occasions Mary even beat John on some test.

     Second, even Mary is deprived by the system of her merited GPA of 2.9. Under the "half-point bonus" scale above, she still was the same as John at 2.8.

What the Case Histories Mean

     What does all this mean and why is it so? It means that many students are unnecessarily cheated out of the credit they deserve by the "system". This results in unnecessary tension for the student who then takes it out on the teacher in particular or the school in general. Parents put added pressure on their children to raise GPA's because they are investing heavily, not only financially but also in other ways as well.

     But it is done this way because it is administratively convenient. Adding up small numbers from 1 to 4 is certainly easier than those from say 50 to 100. That is, it used to be! With the technological revolution that is upon us, it is no longer true. Calculators and computers are changing our lives and our world at an unbelievable pace, everywhere it seems except in the computation of GPA's. There we are clinging to an outmoded, arbitrary evaluation system, or making changes ever so slowly. The adoption of the half-point bonus method is certainly a step in the right direction, but I'm sure John and Mary would prefer to be judged on their percent averages rather than the values obtained by either quality point system.

     The question then becomes why not do it this way? It's not too difficult to do so nowadays with our modern technology. We teachers have always had to do the "hard" arithmetic when asked to compute end- of-year grades in the past. We could do the same with quarter grades and the semester tests, that is, convert to letter grades with their correspondingly assigned quality points. But the effect is the same: lowered values. Of course, we don't do it, and I for one wouldn't want to. But the procedure is being done, across different courses as opposed to across different quarters within a course. And it is the student who suffers most, who is being denied the credit on work he had struggled so hard for.

The New Method Up Close

     Now let's examine the issue a bit more closely, from the standpoint of GPA equivalencies as well as what can happen when grades are changed.

Chart I: GPA-Percent Grade Equivalences
A     100-90     4.0
 883.8 782.8 681.8
 873.7 772.7 671.7
 863.6 762.6 661.6
 853.5 752.5 651.5
 843.4 742.4 641.4
 833.3 732.3 631.3
 823.2 722.2 621.2
 813.1 712.1 611.1
 803.0 702.0 601.0

     This chart is based on the following rationale. When a student receives all 80's, he would have a pure 3.0. But unfortunately, if another were to score all 89's, he too would only have a 3.0 (under the integer system that is still widely used). Yet the person who scores all 90's attains a pure 4.0. This seems statistically unfair. The chart tries to give an interpolation and inter-relation for those raw score averages in GPA equivalencies that truly reflect the demonstrated strength of the student. Failure to do so for the recipient of the higher grades could lead to dissatisfaction for working hard, and on the other end encourage others to go around begging their professors for that extra "fractional" portion just to get their final grade up to that next magical level. And for the guy who works extra hard to get even higher A's, what's in it for him?

     Now, observe these charts.

	   Chart  II-A:   Percent Changes, Integer Scale

		Letters		Points		% Increase

		 B to A		3 to 4		  33 1/3
		 C to B		2 to 3		  50
		 D to C		1 to 2		 100
		 F to D		0 to 1		 "infinity"

	   Chart II-B:    Percent Changes, Half-Point Scale

		Letters		Points		% Increase

		B+ to A		3.5 to 4	14 2/7
		B  to B+	3.0 to 3.5	16 2/3 
		C+ to B		2.5 to 3.0	20
		C  to C+	2.0 to 2.5	25
		D+ to C		1.5 to 2.0	33 1/3
		D  to D+	1.0 to 1.5	50
		F  to D		0   to 1.0	"infinity"

     At first glance, it would seem to be in the student's arithmetic favor to work for (or beg for) that extra point to jump up into the higher category; after all, look at the healthy percent increases in the GPA numbers. And all for the price of "one measly little percentage point!"

     But let's return to John's Chemistry grade and see what can happen with a mere 70 (either earned or "regalado"). His percent average increases to 80.8, not too much by some standards. But his GPA's under the two systems become

   Integer System	Old GPA   2.6	(13/5)
			New GPA   2.8	(15/5)

   Half-Point System	Old GPA   2.8	(14/5)
			New GPA   2.9	(14.5/5)

     Admittedly, the percentages of increase were greater under the GPA calculations (7.7% and 3.6%, respectively) than for the raw score average (0.2%). But I don't think that would impress John too much because the GPA's still lie in the C+ range of things, while his raw score average is still in the B- range.

     The situation is similar, though less dramatic, for Mary. Her likely change in U.S. History, going for a new grade of 75. Under the half-point scale, the only one that gives her the possibility of advantage, her GPA of 2.8 rises to 2.9, while her raw score average only goes to 79.2. Yet she too should prefer the latter because it is still technically better for her in the long run.

* *** *

Comments, Final Questions, and Conclusions

     What does all this boil down to for our students? It's really quite simple: don't use either system of GPA calculations. Instead of setting a minimum GPA of 2.0 for entering freshmen, just say "We'll use your raw score grades, and if your average computes to 70.0, you're in!" After all, it is equivalent to the same 2.0 now being used (as per Chart I), and it gives an advantage to the student, but not anything unfair or inappropriate. It gives full value to any credit given by the teacher, as an 84 is certainly nicer than an 81, and was probably harder to earn as well. And of no little importance to me is that it recognizes the excellence of the high A-student, of which I often have 2 or 3 every year, that otherwise get ignored in the older systems. The administrative work is not any different than at present as the computer doesn't mind if numbers are large or small. It certainly should please the students and parents. And it certainly pleases me, a teacher. It seems to be a bigger step in the right direction.

     There seem to be a few questions that will still need to be addressed, like what about F's, or what about incoming students whose transcripts only show letters. The matter of F's would appear to be a sticky one indeed, for of course a F normally indicates that the individual hasn't met certain standards or achieved certain objectives of the course. In other words, he failed! Well, unfortunately it's not quite so simple as all that. I claim that there has been learning going on in that individual, if the course is a good one, well conceived and managed by by the teacher. I have heard of a student who did not pass a course of mine, yet went to another school and did well there. When asked why he could do a certain type of math problem so well, his reply was, "Well, Mr. Trotter taught me how!" So he was learning after all.

     Here is a realistic case to demonstrate this situation: Billy Black, someone never known for excelling academically.

		Course I	75	C+	2	2.5
		Course II	66	D+	1	1.5
		Course III	71	C	2	2.0
		Course IV	68	D+	1	1.5
		Course V	52	F	0	0

		Averages	66.4		1.2	1.5
By Chart I, Billy's raw score average translates into 1.64, and I'm sure even he can appreciate that difference!

     So perhaps we could even grant some credit recognition for those F raw scores, at least to a certain level. Obviously, they don't help much in raising an average to the 70.0 (=2.0) level, but they likewise don't leave that "goose egg" in the list of GPA numbers. That latter fact of "no points at all" can be very depressing to the learner, as it depresses the GPA rather substantially. And besides this is really an internal matter, the way we wish to govern ourselves locally. When we send grades out to other schools, either for transfers or graduates, we can do one of two things: 1) leave the raw scores with "no comment", allowing the receivers to do what they choose (they probably would anyway); or 2) at that moment include a short statement as to what we considered to constitute an A, a B, etc., and let it go at that.

     Finally, as to those individuals who transfer in from outside our system, first we could try to ascertain officially what the original numerical values were, or at the very least, what the category limits were; they should exist somewhere. If not, then we have to make an educated guess, judgment, as to what would be appropriate and acceptable to all parties concerned. As it stands even now, we accept "outside" letter grades into our GPA scales without being concerned on what raw percentage scores they were based; not all schools use our "decade" system, we well know.

* *** *

     In conclusion, I have attempted to present a rational, reasonable case how one method of computing averages provides various advantages over traditional methods, insofar as they determine grade placement, whether or not one should be placed on academic probation, etc. It's a method which gives every benefit of the doubt to the true achievement record of the student without being unfair to him, his classmates (when it comes to purposes for ranking validictorians, etc.). It also encourages and maintains the credibility of the teacher whe he makes his considered evaluation of work as it was performed by all his students. It should in addition discourage the prevalence of the problem mentioned at the outset of this article. "Puntos regalados" would now be not such an important issue for either student or teacher, though there would always be the case that "a little extra" could make a necessary difference for one's average. No system of evaluation or categorization is ever going to be perfect.

     Most of all, however, it should reduce the tensions and pressures for all parties: student, teacher, parent, and administration. And probably contribute to a healthier academic atmosphere for all, where we can get on with the main purpose of our work: the education of the young people in our charge.


     One final matter remains to be considered, that of how half-credit grades, such as Physical Education, are handled. It is entirely an internal matter for the computer. The portion of the program that computes the average simply reads the raw score for P.E., the divides by 2. All the raw scores are added with the "half-scores" just computed. The divisor that figures the RSA (raw score average) is the sum of the number of full-credit courses plus one-half. It's a simple application of the arithmetic of "weighted averages".

     Here's an example for Susi Smart:

				     Computer  Int.     H-P	

	Course I	93	A	93	4	4.0
	Course II	88	B+	88	3	3.5
	Course III	82	B	82	3	3.0
	Course IV	79	C	79	2	2.5
	Course V	83	B	83	3	3.0
	 P.E.		96*	A*	48	2	2.0

	Totals			       473     17      18.0

	Averages			86.0    3.09    3.27	

     Once again we see the statistical imbalance of the traditional GPA systems. Susi's translated GPA (new style) would now be 3.60. (Incidentally, her A in P.E. raised her RSA of her core courses from 85 to 86, a fact that Susi would certainly approve.)

     Finally, it would be in order here to observe the most important fact in all this debate. The RSA will always come out higher than either form of the regular GPA, eventhough sometimes ever so slightly. [The only exception to this occurs when ALL of the grades fall on the lower limit of the range.] This is due to the simple mathematical fact that within any given GP range the student's raw scores are always being shifted downward to the lower limit of the range. In effect, this reduces, for example in the half-point scale, all 79's, 78's, 77's, and 76's down to a 75! And it's even worse under the integer scale. Poor student!


This item was written in May 1988, merely as a personal catharsis, during a very troubled end-of-school-year period in my career. It is given here for what it is worth, little or much, take your pick.


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