Friday, April 19, 2013

The Problem with Averaging Grades

It is common practice for teachers to calculate grades for their students by "averaging." That is, teachers add up the points students have been assigned for their work, and divide by the number of points possible. Mathematically, that's what "averaging" is--computing the mean for a data set.

For example, imagine that a student received the following marks on assignments throughout the term:

92, 84, 87, 60, 88, 89

Calculating the mean (average) would entail adding up these numbers:

92 + 84 + 87 + 60 + 88 + 89 = 500

And then divide the sum by 6 (because there are six items in the list):

500 ÷ 6 ≈ 83.3

Which means the student's grade would be an average of 83% when written as a percentage.

If you are a mathematician, you will be nodding by now and saying, "Ummm...yep. Your point is...?"

But here's the thing--I think that kind of "averaging" may not accurately reflect a student's learning. Think of the assignments above. Even if you buy into the idea that a collection of points on assignments is an accurate way to measure a student's learning (I'm beginning to really doubt this, by the way), I'm not sure that the 83.3% really captures the true learning. That one "60" brought the average down--probably a whole step on the grading scale, if not more. Does that one score diminish the rest of the learning from the term?

Maybe it's clearer if we discuss really low scores. What if the "60" was a "32" instead? Now the situation would result in an average of 78.6%. Does that seem right? When all the other scores for the term were centered in the upper 80s?

Or how about a zero? What if the "60" were changed to a "0"? (Maybe the student didn't turn in this assignment at all?) Suddenly our student's average has plummeted to 73.3%!

Now, here's the thing--if you would throw out the low score (whether it's a 60, or a 32, or a 0), this student's average would be an 88%.

Thinking over the whole term's scores, what seems to be the truest measure of what our student has learned? 88%? 73%?

And let's really think about what that percentage represents. In the situation resulting in the 73%...does that mean our student has learned 73% of the material taught this term? If our student had turned in that missing assignment--the one we scored as "0"--would that mean our student has actually learned 88% of the material instead?

I think there are four real problems with the Bucket-O-Points method of averaging grades:

  1. Averaging grades this way can turn our students into grade-grubbers, fighting for and quibbling over every point. Rather than focusing on whether or not they are learning, they are focused on what they are earning. And don't get me started on extra my experience, it's not the kids who could really use the extra points who come looking for's the kid with a 98% who really wants the straight "A." (I have other concerns with extra credit as well...not really the point of this post...maybe another time.)
  2. Averaging grades this way can turn evaluation something punitive: on the teacher's end, we "punish" students for not turning in work or for doing incomplete work. In my mind, this doesn't really show what students have learned. The meaning of the grade is distorted, and becomes a measure of compliance, rather than a summary of learning.
  3. Averaging grades this way can make teachers seek to assess what is easily quantifiable (factual knowledge, right-or-wrong answers) over what is most valued (holistic understanding, depth of thinking.) I'm not suggesting that student shouldn't "know"things. But I hope we are more interested in how students put the things they know together: how they apply, analyze, synthesize, and evaluate ideas.
  4. Averaging grades this way seeks to reduce the richness of a student's learning to a single symbol. I think this is a culture-wide problem, not just one in schools. We love to quantify and rank and compare and an average grade allows teachers to do this more easily. The problem I see in this is that this kind of reductionism has to leave things out. A portfolio of artifacts is--of course!--going to be a more authentic means of demonstrating a student's learning than a single symbol. But we have so much cultural inertia behind the quantifying...
I think that we educators need to give our assessment practices a serious, honest look. Does "averaging" really give an accurate picture of students' learning? And if not, how shall we proceed?

Students, parents, and fellow educators, I'd love to hear your feedback and reactions to these ideas!


A bit of a postscript on this topic...

To be fair, I'm still struggling with this in my own teaching practice. Back in 2007 as a middle school teacher, I began using standards-based grading practices in my science and computers & media classes. Now that I've moved into higher education, I've swapped back into a sort of Bucket-O-Points method. I'm really wrestling with this right now--I continually strive to reduce the gap between my philosophy of education and my actual classroom practice, and I see this as a big problem right now.


  1. thank you dave, for so much to think about (again!)

    ah, the 'throw out the lowest grade' teachers are well liked. i know of students who will deliberately fail or not show up for their last assignment, since they know they will still get their A in the class... there are others who truly need to master the concepts of the 'failed or low grade assignment/test' in order to be successful later on, when those concepts are used for higher learning. the challenge for those that have low scores to prove mastery in a different venue (ie a self directed paper, or project) that shows mastery of the concepts seems to be a fair way for them to be able to redeem that low grade so the average goes up! (BUT...more work for the teacher)

    but...i digress...

    the report cards from my parents were on a '10' scale (with no fractions). this seems more accurate. i especially appreciated seeing theirs (and others) where it was not straight across the board 10s....

    thank you! :)

    1. I appreciate the feedback, Grace. I know what you mean about simply throwing out the low score--and that's not really what I'm advocating for as an across-the-board policy. (Though as I re-read this post, I do sound like that's my argument.) I would suggest though, in some cases, that dropping a low score would be a fairer representation of a student's learning.

      I really think assessment needs to be about uncovering what students have learned: what they know, understand, and are able to do after a lesson or unit that they didn't know, understand, or have the capacity to do beforehand. Grading is just the reporting part of the process, and that's the part I think is most broken in my own teaching practice and needs to be addressed.

      I love your point about providing opportunities to demonstrate learning in a different venue--THAT is what I'm really thinking of too! If it's a goal we set as a learning target for students, it's important that we assess it. And if they haven't learned it (yet), they should have another opportunity to show that they have mastered the content. Redo's and retakes are another whole conversation--I have a post in mind on this topic, but it'll take me a bit to collate my thoughts about it.

      In truth, this series of posts on grading has been rolling around since last fall...and I've finally gotten up the nerve to shape them up and post them. Sort of scary for me, because I have a feeling I'm stepping on toes...

    2. ah yes, sometimes a student doesn't perform well on a 'test' or assignment for different reasons - when IS the perfect time to test each student - some have poor testing skills (but that is a whole different subject eh!) it probably is a better (fairer) representation of their general knowledge....

      sometimes...although this wasn't addressed in this blog, the testing can show a (dare i say this) a lack of effective teaching for a particular concept. sad but true... sometimes (some) teachers have the lowest grade thrown out to fix this problem...of their own making... again, a whole other blog posting!

      do overs and re-takes, yes, a whole other subject...

      and, i hope you will address the issue of teaching for the test....(especially for the standardized tests) in the future..

      i appreciate that you are thinking of these issues, and dare to write about is an interesting subject (well, all of teaching is interesting...)

      thank you!

  2. I agree with most of what you said and think that a single grade is a poor way to evaluate students too but we do give tests and assessments all year long and they get single grades on those over certain topics. Unfortunately I don't think we will ever convince enough people or find the perfect solution to a single grade. I also think that we want to make sure that students know what they are having trouble with and know the importance in a class of understanding all of the material. When I was in college I got A's on all of the weekly tests except one that I flunked. I had a hard time with the topic that week. I got a B+ in the class. The grade was deserved. I didn't deserve an A because I didn't understand a part of the calculous class. I wouldn't feel confident teaching that part of calculous either so don't think I should have gotten an A as I only had a B+ worth of knowledge on the topic. In my math class today I stress learning the material not grades. I give five points for each daily assignment- 5=A, 4=B, 3=C, 2=D, 1=F. I allow the students to compare answers every day before we check their assignments and help each other with things they are having trouble with. My goal is for them to know the material by test time. My tests are worth 100. It has worked well and students seem to understand the importance of learning not just competing for a grade.