What grade is an “F”?

In Good Math, Bad Math, Mark Chu-Carroll talks about a USA Today article about grading systems in schools. The article focuses on arguments about what an “F” means, and the “fairness” (or not) of systems that assign “A” to scores of 90-100, “B” to 80-89, “C” to 70-79, “D” to 60-69, and... “F” to 0-59. How can it be right, those who would change the system say, to have 10 points each assigned to the other grades, and a whopping 60 points assigned to “F”? [Update to clarify: the quotes below come from the USA Today article, not from Mark's blog.]

In most math problems, zero would never be confused with 50, but a handful of schools nationwide have set off an emotional academic debate by giving minimum scores of 50 for students who fail.

Officials in schools from Las Vegas to Dallas to Port Byron, N.Y., have proposed or implemented versions of such a policy, with varying results.

Their argument: Other letter grades — A, B, C and D — are broken down in increments of 10 from 60 to 100, but there is a 59-point spread between D and F, a gap that can often make it mathematically impossible for some failing students to ever catch up.

Mathematically impossible? Say what?

Mark addresses the issues nicely, pointing out that there might be valid arguments for a change, depending upon exactly what they’re trying to do — the basic problem is in converting from percentage scores to letter grades and back. There are a couple of things I want to say beyond what Mark said.

I want to highlight a discussion in the comments section of Mark’s post: for most of the sorts of exams we’re talking about, there’s some sort of “baseline” score, a score that one could expect to get by random chance, even if one knew absolutely nothing about the subject. On a true/false test, 50% is that baseline, a grade that could be achieved by a pre-school child who randomly filled in answers.[1] On a multiple-choice test with four choices for each question, the baseline is 25%.

So there’s a perfectly reasonable argument that the baseline score, the score that could be expected of an entirely ignorant student, should be where the “F” grades start. You should have prove you know something in order to get even a “D”.

And, now, what’s that about “mathematically impossible”? Here’s some clarification:

“It’s a classic mathematical dilemma: that the students have a six times greater chance of getting an F,” says Douglas Reeves, founder of The Leadership and Learning Center, a Colorado-based educational think tank who has written on the topic. “The statistical tweak of saying the F is now 50 instead of zero is a tiny part of how we can have better grading practices to encourage student performance.”

And that statement, itself, is a glaringly good example of a classic mathematical dilemma. This is not a statistical problem, and trying to apply statistics to it is silly. Students do not have “a six times greater chance of getting an F,” unless they are playing the random-chance game with the test, trying to beat the baseline score with guessing and odds alone (and even then, it’s not six times). And changing the grading system is not a “statistical tweak”.

The tests are measuring (or trying to) whether you know the material. What Mr Reeves is saying is that your knowledge combines with some roll-of-the-dice chance to create your score, and that even if you learn the material, you have the odds against you because of the scoring bias. What I’m saying is that knowing the material, not random chance, is what you need to pass the test.

Then, too, is the question of fairness. Excuse me: exams are not meant to be fair, in the sense of distributing grades equally, or of giving students who haven’t learned the material an equal chance to those who have. Someone who can correctly multiply two three-digit numbers 50% of the time is clearly better off than someone who can only succeed 10% of the time, or not at all. Yet we still might consider 50% to be failure at that exercise.

Suppose we decide that to measure performance for the football team tryouts, we should divide students’ performance in doing push-ups in groups of 10. If you can do 90 or more, you get an “A”. 80 to 89 is a “B”; 70 to 79, a “C”, and 60 to 69, a “D”. If you can’t do 60, you get an “F”. We’re grading it that way because we’re using it for qualification for the football team: if you don’t fail, you can be on the team. Would anyone think it “unfair” that 60% of the data points denote failure?

Of course not. We would say that we require that our football players be able to do at least 60 push-ups. We would say that we have minimum standards. We wouldn’t say that every kid should have an equal chance to be on the team — only that each should have an equal chance to try out. Same with academics. We have minimum standards, and it’s not unfair to say that when you don’t meet that minimum, you fail.

There is the issue that a zero (or a score close to it) on one test, when put into the overall grade formula with the rest of the semester’s work, will be hard to overcome. That’s true, but it would be a unusually hard-nosed teacher who would look at one grade of, say, 10, mixed with other grades of 65 or 70 or so, along with demonstrated effort... and wouldn’t allow some sort of make-up work to replace the inordinately low score.

We don’t need to revamp the grading system. We need to have students who take the work and the learning seriously, and teachers who look at the whole picture and act (and grade) accordingly.

We have a tendency to want to stuff everything into a formula, to come out with a well defined answer. Life is not like that.
 

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[1] In fact, it would be as hard to get a zero on a true/false test as it would be to get a 100. I’d almost be inclined to give an “A” to someone who got none of the answers right, on the assumption that it was done as a joke, because you’d have to know the material quite well in order to manage to do that badly.