## Question 7

### Most teachers assign final grades by using the mathematical mean (the “average”) to determine them. Give at least 2 reasons why the mean may not be the best measure of achievement by explaining what the mean hides.

All measures of center hide variation.

This is what makes them useful, and it is what makes them problematic.

Using the mean makes zeroes a problem in grading. Wildly divergent values (such as a zero in a gradebook) will greatly affect the mean. It is hard to argue that 2 A’s and a zero is the same as consistent D work. Yet this is how the mean plays out.

But going too far down this road will only lead to critiques of the whole system of grading students at all. I find that system to be indefensible and counterproductive. I have made my peace with it, and I try to do as little harm as possible with the responsibility I have to assign grades in my work.

All of which is to say, it is not* using the mean *that leads to a poor measure of achievement. It is *mistaking quantitative measures for accurate ones* that leads to a poor measure of achievement.

Yes, I have serious problems with the very idea of grading, even in mathematics. That said, the mean masks variability while at the same time being particularly volatile. For small n, it is absurdly susceptible to the effect of outliers. Median and mode, while each having weaknesses of their own, are more robust in that regard.

But then there is a phenomenon I experienced only in upper-division undergraduate mathematics courses at U of Michigan: heavily curved grading. For example, my elementary number theory class was such that not only were the exams (a midterm and a final) curved enough that scoring in the low 60s out of 100% was considered outstandingly good, but the professor (a Englishman who was teaching in a post-doctoral position) made clear that he was very pleased with the scores which averaged in the high 30s, if memory serves. I can’t imagine being happy with a class averaging, say, 38 on an exam. He said that the results were typical. I wondered whether that was a comment on his teaching, the subject, the way the course was structured, something about the nature of the exams, a reflection on math majors and other majors in math-intensive subjects, or some other factor(s).

Zero scores are a big problem with using the mean. To deal with that, I change every 0 to a 40 at the end of the semester. (One notch below the 50-64% I use for a D, with 65-79% as a C.)

Another problem with the mean is that it isn’t reflecting *when* the grade was earned. I allow test retakes, so students can improve their grade. I also give a comprehensive final, which is harder for students to do well on than the unit tests. If a student gets a better grade on that than their mean, that becomes their grade.

I teach in the International Baccalaureate Middle Years Program, and we have a very different system of grading, which I love. I think many of you would like some of its ideas as well, so I’ll try to quickly summarize them.

First of all, students are graded on four criteria: A- knowledge & understanding (tests), B- investigating patterns (investigations), C- communication (reports or investigations), D- was reflection but is now becoming applications (real world problems). Everything is graded through the use of a rubric, with the highest score for any given criterion being an 8.

When I grade a test, a student does not have to have everything correct to get an 8, they just have to show overall mastery, including completing unfamiliar problems which are meant to extend their knowledge to types of problems never seen in the class. If they show mastery but cannot do unfamiliar problems, they earn a 7 (still an excellent score). Levels 5-6 say that a student can complete more challenging problems including in applications. A 4 is the lowest passing grade. The great thing about grading with a rubric is that if a student makes a silly mistake on an easy problem, it doesn’t affect their grade at all. Assigning points to problems and taking an average is explicitly not allowed.

But the thing that hits most on what you’re talking about here is how you calculate the final grade. For each criteria, you must decide what the student’s best sustained effort is. So for example, if on the first test the student really bombs and gets a 2, but subsequently gets a 4, then a 6 and another 6, it’s as if the 2 (or the 4) never occurred. The student has shown that she can maintain a level 6, thus that is the grade she receives. In that way, we encourage growth.

The last piece of the puzzle is to total up the grade from each criteria and use grade boundaries to determine the final grade. This has been helpful for students who struggle to perform in a test but can write a beautifully detailed and thoughtful report.

Anyway, long comment, but thought you’d find the idea of “best sustained effort” a really nice approach.

Only 1 problem has been mentioned – the zero. Yes, the IB system is better than the averaging into a single grade!

But I asked for (at least) two problems with the mean. One other problem is that the mean hides variation, so that a student who gets ACACADCA gets the same grade as BBBBBB; which seems wrong. A third problem is that progress is not rewarded but it should be, presumably.

OOPS – made an error. You DID mention variability, briefly. The ‘third’ one still stands, too.

I love the story of a bicycle riding course, where a student falls (fails) on the first quiz, and the second, and the third… but at the end of the course, the student navigates the obstacle course perfectly and receives an A: their first and only non-zero grade. What should their grade be? If the purpose of the grade is to:

– document their current skill level, then it should be an A

– document their improvement, it should be an A

– demonstrate their effort, it should perhaps also be an A

– reflect their average skill level: it is an F

My post/thoughts on the 0-100 grading system (and averaging) : http://mathmaine.wordpress.com/2013/01/17/unintended-consequences-of-a-0-100-grading-system/

If learning is really the goal, why allow zeroes anyway (may be part of your “but going too far down this road will only lead to critiques of the whole system of grading students at all”)? In my classroom, individuals work until everything we do is mastered…so grades mean very little because my idea of an “A” is that a student has shown mastery for each concept no matter how long it takes. This may not work as well in the college setting with grades being more important (maybe/maybe not???), but I think it works well with middle school students because they always have the goal of improvement. This is a great post for teachers to refer to as they start the new school year! Thanks!