Grading and the 'extra credit' device

The results of the semester’s third test have just been returned to the students. There are only three more lectures, a short break, and then the final examination to go. Student Alan is on track for a B-plus letter grade, and is out of range for an A-minus grade. After the lecture, Alan speaks to the lecturer.

Alan: “Sir, I see that I am out of range for an A-minus grade. Can I do something for extra credit?”

Lecturer: “Yes, you are out of range for an A-minus grade. You can still aim for a B-plus grade.”

“I am worried about my GPA and graduate school. I need extra credit.”

“How many marks would you like the extra credit to be worth?”

“Fifteen marks will be enough to get me into the A bracket.”

“Yes, I can set an extra test worth 15 marks. It will cover all topics to date. We conduct the extra test during consultation time immediately following the second lesson from today.”

“Sir, that’s not fair. It should cover only the topics in Test Three.”

“You’ve been losing marks since Test One. That’s how you became out of range of an A-minus grade. If the extra test covers only Test Three topics, then you’re essentially re-taking Test Three. The new test would be only a supplementary Test Three, and the score will replace (not add to) your Test Three score.”

“That will not give me enough marks to get an A grade.”

“Well, you do your own calculations. Another thing: I must offer the same extra test to all the other students – in fairness to them. Of course, they can opt to stay with the marks they have, and not take the extra test.”

“That’s for them to decide. I’m interested only in my extra credit.”

“There’s one more thing. With the extra test, your letter grade will be computed by taking your total marks (including from the final examination) as a percentage of 115 marks, instead of as a percentage of 100 marks.”

“What? No! That’s not fair! I will likely be out of range of A again! You add marks only to the numerator, not the denominator.”

“Alan, your misunderstanding is quite common. Grades are awarded on the basis of the percentage of offered marks that you earn, not on the basis of the number of marks you earn. This common misunderstanding is caused by the common practice of using a 100-mark scale in assessment, when percentages are calculated also on a 100-point scale. People (students, lecturers, administrators) conflate the two. If grade computation uses only the numerator, then a student who earns 100 out of 1,000 offered marks must be awarded an A-plus letter grade, and be considered “High Distinction”. But this is clearly absurd, since the student has mastered only ten percent of the course material. Consequently, students who take the extra test will have their letter grades awarded on the basis of their percentage of 115 marks, whilst students who do not take the extra test will have their letter grades awarded on the basis of their percentage of 100 marks, in the usual way.”

“None of our other lecturers does extra credit this way. I will be speaking to the student union about this, and about you.”

“So do you want the extra test or not?”

“I want the extra test. And I want the denominator to remain at 100 marks.”

END

To not bell curve or to bell curve grades

To not bell curve or to bell curve grades

In May 2019, Singapore Management University decided to review the letter grades of all 169 students in a class – because the lecturer had awarded every student an A grade, despite the university’s rule to limit the number of As in any one class to a third of the class. On 17 June 2019, The Straits Times ran a feature article on the non-use and use of bell curve grading in Singapore universities.

In this article, I would like apply some critical thought to grading on a bell curve.

We must first establish the purpose of grading. It is to inform some third party (not the lecturer or student) how competent the student has become in the subject at the end of the course. The most likely third party is the student’s future prospective employer. I exclude the lecturer because he or she already knows the answer from class interaction and formative assessments (class exercises, homework, and other non-credit work). The student also knows the answer from the above and from self-reflection.

Hence, the bottom line in discussing grading method is what information it provides to that third party, most likely the student’s future prospective employer. We return to this at the end of this article.

Typically, a student’s competence in the subject is measured through various modes such as quizzes, tests, presentations, projects, essays and examinations. Various weights are assigned to each mode, with the total amounting to 100 marks. That is to say, students are measured on a unique 100-point scale.

This is unlike such scales as for temperature (Celcius, Fahrenheit, Kelvin), length (imperial, metric), electrical (volt, ampere, ohm). A measure of 300 degrees Centigrade means exactly the same thing to a physicist, meteorologist, doctor, or indeed anyone. In contrast, one mark in any given course means one mark in only that course. Each academic measuring scale is unique to the course in which it is used.

At the end of each module, each student would have earned some number of marks for that module. Let’s call this the numerical mark.

The institution, faculty, department or lecturer (the decision-maker varies) sets a minimum number of marks as the pass mark. This establishes the minimum point beyond which the student is said to be competent in the subject. Typically, this minimum varies between 35 and 60 marks. Beyond this, we have a 65-40 point scale of competence.

This is the point where grading kicks in.

Grading is typically not reported on such a multi-point scale to the third party. The typical scale used is the letter grade scale, ranging from A-plus to D-minus – a twelve point scale. The pass mark range is typically equally divided into twelve parts, each corresponding to a letter grade. Each student is awarded the letter grade into which his or her numerical mark falls.

We need to collapse a 65-40 mark pass range into a 12-point letter grade range. Dividing equally, each letter grade covers anything between 5.4167 (65 ÷ 12) and 3.3333 (40 ÷ 12) marks.

Here is a simple example. We set the pass mark at 40 marks. That yields a 60-mark pass range. Dividing this into 12 parts gives us five marks per letter grade. A student scoring 88 marks would get the letter grade “A-“.

This is grading without using a bell curve.

The term “bell curve” is we commonly call the “normal distribution” (which when graphically expressed exhibits a bell shape). This is the distribution of any given feature (such as height, weight, intelligence) in an unbiased population. The caveat is that the measuring instrument must be sensitive enough to distinguish small differences in the occurrence of the given feature. For example, it is good enough to time marathons to the nearest second, but 100-metre races must be timed to the hundredths of a second.

In the same way, a well designed academic measuring scale will uncover student competence in a normal distribution or bell curve shape. This result may be taken as the criterion of good test design. (I once achieved a perfect bell curve with a class of 17 students).

As mentioned above, Singapore Management University set a rule for no more than one third of a class being awarded an A grade. This is a maximum. What happens if more than a third of the class achieves 85 marks or more (to continue our simple example)? What happens if no student achieves at least 85 marks?

This is where the bell curve method of grading would kick in – if mandated.

Percentages (not necessarily equal) are assigned to each letter grade or grade group (B-, B and B+ grades etc.) and students are awarded a letter grade as they progressively fulfil the percentage in each letter grade or grade group.

Bell curve grading is essentially a method to award grades according to a student’s rank (colloquially known as position) in the class.

We can now answer the question: What information does a letter grade provide to the third party, most likely the student’s future prospective employer?

If the letter grade was awarded based on the student’s numerical mark (that is, no bell curve was applied), the letter grade represents the student’s achieved competence relative to the academic measuring instrument / scale used in that course or module.

If the letter grade was awarded based on the student’s rank in class (that is, a bell curve was applied), the letter grade represents the student’s achieved competence relative to the other students in that course or module.

But the basis of awarding the letter grade is invisible to the third party.

Hence, a student showing an “A” for a given course in his or her academic transcript must be understood to be a student who was EITHER a student who achieved top competence relative to the academic measuring instrument / scale used in that course or module OR a student who achieved top competence relative to the other students in that course or module.

This is as good as it gets. The prospective employer must also use other criteria to assess whether it should hire that job applicant.

A final comment: It would be much more informative to the third party to report BOTH a student’s numerical mark (eg. 76 marks) AND the student’s class position (eg. 11 / 17).

Cheers.