Thrown a curve

(“Thrown a curve” is a phrase from baseball, meaning when someone throws you a curve ball that is difficult to hit; it can also mean running into something unexpected.)

Halfway through the semester of Gen Chem I, we had just gotten another exam back, and things were grim. On the first day of class, the prof had told us that, “Half of you are going to drop out or flunk,” and he hadn’t been kidding; as we neared the last day to Withdraw from class, the students were dropping like flies. Those of us still remaining were struggling mightily. The students were bitching about the teacher, and in turn the teacher was complaining about “the kind of students nowadays” (and this was back in the early 1980’s).

Of the several dozen who hadn’t given up and were slumped through the lecture hall staring at their exams dripping with red ink, only two had done well, meaning had correctly answered at least 70% of the questions. (Hallway discussions after lecture would yield the fact that both of them had taken chemistry in high school, so this wasn’t their first experience with the concepts.) As the instructor skimmed through and told us the correct answers to the test, the grousing turned to arguing, and then to deal-making.

“Do you grade on the curve?” pleaded one student. Everyone turned expectantly towards the prof, who as usual, looked annoyed and cross. His utter fatigue with teaching had been apparent from the first week, and had disimproved steadily with the succeeding weeks. His answer, like all other quantitative answers, began with a sigh audible all the way to the back of the lecture hall, and then he rambled on in a rush of words as to how such a calculation would work, and then why it wouldn’t change anything on today’s exam because of the two students’ A and B grades in the 90+ and 80+ percentiles. After giving them an earful of arithmetic, the energy of the protesters was worn down, and he returned to reciting the answers we should have gotten. Why we had not gotten them was not an issue he discussed.

Later on that day I was more puzzled by grading curves than by acid-base reactions. (The conceptual part of chemistry was fine, I had simply gotten tangled up in the calculations. Again.) Not yet having the awesomeness of the World Wide Web for looking things up, I flipped through some maths books at the library until I found mention of the Normal Distribution Curve in a statistics book.

I understood grading by percentile; a score greater than or equal to 90% was an A, 80% was a B, and so on. And I understood how the normal distribution curve worked as far as describing how most of the members in a set were in the middle range, and successively fewer were at the lower and higher ranges. But trying to apply that normal curve (a mound that looked like a sand dune, or slice of bologna after my dad had cooked it in the pan) to distribution of grades left my brain itchy.

Everyone knew that a C grade was “average”, and that C’s were common, and A’s and F’s were rare. That should then mean that the Normal Distribution Curve was being supported as a pedagogical concept. But something didn’t seem right. I figured that “mental itch” feeling meant there was something wrong with my understanding; after all, it was obvious that I had major problems with calculations.

In later years I studied statistics, and learned that not every data set would follow a normal distribution curve. Some of them followed asymmetric curves with their central tendencies over to one side or the other, some of them were two-humped (the Bactrian camels of statistics), and some data sets didn’t make any particular sort of curve at all. I also learned about statistical circular arguments, whereby creating a measurement algorithm that would result in survey scores with a normal distribution curve did not prove that a population set naturally fell into such a curve — the curve was simply an artifice of the algorithm.

I have since learned that the “mental itch” feeling does not necessarily mean I am being stupid; more often it means that something else is Not Right.

Weird things happen when people try to force students’ grade into the curve. It’s not that the scores cannot fall into a curve. Rather, it’s that people try to use curves when they shouldn’t.

With the standard grading scheme, a student has to achieve a certain percentage to be considered as having mastered whatever was being assessed. (Whether or not that assessment accurately reflects the learning objectives is a whole ‘nother story.) But if we instead impose the normal distribution curve to sort out the A, B, C, D and F grades, we then say that the top grades are A’s, the bottom grades are F’s, and the median (and frequently mode) grades are C’s. There are a couple of problem with this. Firstly, it requires that some students get bad grades. Secondly, the distribution of letter grades from the curve does not guarantee that the students are succeeding in meeting the required competencies.

In addition to the problems that can be created by imposing curves, we have an essential problem in assuming that grades should even result in a normal distribution curve. There’s that algorithmic artifice issue, where exams can be created that will (when given to a large number of students) result in a grade distribution that creates a normal curve. This is the rationale for the argument for using grade curves. But it’s a circular argument, because not all assessment methods will yield such score scatters, and they should not have the normal distribution curve imposed upon them.

Furthermore, we have to ask ourselves if demanding a normal distribution curve really reflects our educational goals. Do we really want to have certain percentages of students getting bad or mediocre grades? When we ask individual teachers what they want for their students, none of them say that they want lots of average students, a few really good ones, and a few really poor ones. When we read the mission statements for school districts, we find that every district has Lake Wobegon dreams, where they want all their students to be “above average”.

Another concern people have is with “grade inflation”. Because of the pedagogical bias or expectation that grades “should” fall into that fabulous normal distribution curve, when we get lots of students getting B’s and A’s (and hardly, if any, getting D’s and F’s), then people start fretting that something is terribly wrong. Why, there must be grade inflation going on. Obviously, if so many students are getting good grades, then that must mean that the work is too easy!

On the other hand, if most of our students are not only passing tests and courses, but are even doing very well, maybe that just means that the teachers and students are both succeeding in their educational goals. Don’t we want all of our students to pass subjects and succeed? Education is not a zero-sum game, where every winner must be accompanied by a loser. Likewise, if most of the students are doing very poorly, it does not necessarily mean the students are just lazy or stupid.

7 Comments

  1. proteus123 said,

    16 April 2008 at 16:15

    The underlying problem is that while a normal distribution curve may generally fit the grades that large samples of students get, the curve will always look slightly different, assignment to assignment, class to class. A curve is meant to DISPLAY data that is already collected, not be a static line to place students’ grades on. I would call this preemptive statistics.

    It is perfectly feasable to have an assignment that a majority of students perform well on, placing the peak of the curve in say the upper B and A range, and also feasable to have an assignment that perhaps doesn’t relate much to the curriculum, where you would expect a pretty even distribution across the grade range (a better display of this distribution might be a bar graph or even a scatter plot).

    Doing this sort of grade mapping is a great way to guage how well a set of students are doing and can be indicators of how well the teacher is teaching and whether the curriculum is too hard/easy, but as has already been said time and time again, we want students to BREAK the normal distribution curve. We should let them.

  2. qw88nb88 said,

    10 April 2008 at 14:17

    Tracy,

    I think very few teachers want to fail students, although the concept of so-called “flunk out classes” would fit the bill. (If colleges are really concerned about students having the necessary expertise, it would be quicker, cheaper, and a better use of everyone’s time and energy to require placement exams prior to enrolling in particular classes.)

    The situation you’re describing is two different things. One is simply doing well on a mastery exam, and we all want the students to have good understanding of the material. The other thing is, as you pointed out, recognising exceptional achievement.

    A letter grading system determined by cumulative quantitative scores simply does not have the built-in means of expressing the differences between someone who does well through technical mastery, and someone who does well because of their insight and ability to apply critical thinking.

    What can sort out the difference between those is the type of assessments an instructor uses, which for higher scores would require that a student demonstrated work at higher taxonomic levels.

    Sorry if that sounds a little circular, but it’s up to the instructor to decide what kinds of things are important in the student learning objectives, and how those are reflected in the assessment and grading system. Good instructors make this process transparent at the beginning of the semester, and encourage students to increase the depth of their learning.

    andrea

  3. Tracy W said,

    10 April 2008 at 13:23

    Because of the pedagogical bias or expectation that grades “should” fall into that fabulous normal distribution curve, when we get lots of students getting B’s and A’s (and hardly, if any, getting D’s and F’s), then people start fretting that something is terribly wrong.

    Is this feeling due to a desire to fail students, or is it due to a desire that the really good results should be recognised? In any normal class there is a variation in ability. If lots of students are getting A’s, then there’s no particular recognition of the truly great work, unless you have some other way of recognising it.

  4. 9 April 2008 at 13:47

    I don’t grade on a curve, but a strict percentile system, with no rounding. Having said that, with anywhere from 900 – 1200 students, grades distribute normally, as they will when you have a large enough sample space.

  5. qw88nb88 said,

    9 April 2008 at 12:18

    See this post about the Kathleen Seidel story.
    andrea

  6. 9 April 2008 at 1:56

    […] Also 2:  Students who did well in a chemistry class that Andrea uses to criticize the notion of grading on a curve […]

  7. athenivandx said,

    7 April 2008 at 19:39

    okay, this is TOTALLY unrelated to the content of your post, but Kathleen Seidel, an autism rights blogger, is facing a subpoena of alot of her correspondence with other bloggers as well as more private information, because she spoke out against something or other that I cannot remember right now, but Orac of Respectful Insolence has written an open letter to David Kirby and Dan Olmstead, two prominent journalists, urging them to speak out against the subpoena. I have linked to the open letter in my lastest blog entry, and I am urging everyone reading it to link to it and help spread it around the net. The more it spreads, the better are the chances of it being picked up by someone or people with power on the right side of autism rights, and coming out in defence of Ms. Seidel

    thanks

    The Integral of athenivanidx, in defence of Kathleen Seidel


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