Thursday, June 09, 2011

Perpendicular Thinking

As I contemplate this past academic year of teaching, I feel pretty good about my classes (and students!), although, as usual, I have some ideas for things I want to change next time. In the end, through all the ups and downs, it was a good year.

And yet.. the perfectionist in me wonders why I can never get the absolutely key, essential, critical, first-order, important concepts through to every single student such that they can demonstrate mastery of these concepts on the final exam. This year, there was one student who somehow made it through 3 courses (1.5 taught by me) that used a particular fundamental concept in many different ways and was unable to show any knowledge of this concept. Do I need to draw it for you? OK:


I have been teaching for >20 years, and the fact that a Science major came up with D as an answer instead of C or A/B shocked me.

Possible explanations for the student's bizarre wrong answer:

- I wrote an ambiguous, poorly worded exam question. Evidence against this: every other student in the class got this part of the question right, and there is no possibility that configuration D is ever correct, no matter how poorly worded the question.

- The student is "intellectually challenged" and doesn't understand even basic concepts that have been presented repeatedly during a year of classes by different instructors. This is not a very convincing explanation and, moreover, is not supported by other evidence for the student's abilities in these courses.

- The student was rushing, was extremely careless, didn't read the question, just jotted something down to make marks on the paper. Maybe, but it's hard to imagine any level of carelessness that could lead to answer D.

- The student was cheating, but did not cheat well. That is, the student glanced at another student's exam, but got the questions and answers mixed up.

No, I don't believe that. I don't believe the student cheated, but even if I am wrong about that, answer D is not even close to any answer to any question on the exam.

- The student knew the answer but experienced exam-stress brain-freeze. Perhaps in a calm moment, the student could easily have produced the correct answer, or at least not a totally wrong, physically impossible, not-even-worth-partial-credit-it's-so-wrong answer. During the exam, however, the student wasn't thinking straight and, for this question, gave what was perhaps the most incorrect answer possible other than leaving the answer entirely blank.

I do not totally reject that possibility, but the concept in question is not complex. The brain-freeze would have had to have been catastrophic. And yet, the only explanation I find even somewhat plausible is that, under stress, the student confused the elementary concept with another concept that was also discussed in these classes. Out of stress or carelessness, the student described this other concept (but even that not correctly).

I suppose a take-home exam with liberal time allowance is a way to alleviate exam-stress of this sort, but I have tried that format in the (distant) past, and know that in this particular course, students prefer to study, take a final exam, and be done. Fortunately, in this class there are many non-exam graded activities so that all but the most severe case of exam-stress doesn't result in a dire grade situation. [And of course, students with documented learning disabilities can take the exam in a testing center, in a quiet room, and in some cases are given extra time to complete the exam.]

When I think back on the most recent class that I taught, the incident described here is one of the things that sticks out most for me. I'm not sure why, except that even professors who have been teaching for a long time may wonder (especially after grading final exams): Didn't I teach them anything? (Or the variant: Didn't they learn anything?).

The answer may well be yes, but some of us are affected by the spectacular outliers: the exam answers that are so bad, they shock us. What does it mean that I am still shockable after all these years of teaching? That I haven't seen it all yet? That there are still teaching adventures -- good and bad -- to be experienced? I hope so.

28 comments:

Brad Holden said...

Eons ago, when I was an undergrad, I had a final in an advanced course with a couple of easy warm questions. Once had the correct answer of 2 times pi. I diligently computed 4 pi.

Sometimes people just blow through easy familiar material on their way to the more challenging stuff, and get the wrong answer in the process.

Isabella said...

As somebody how used to suffer from extreme brain freezes (One time, I knew the questions and the answers before the exam, and couldn't answer any), I would believe the brain-freeze hypothesis.

FatBigot said...

Can you simply not ask the student?

"This is a foundation concept thay you need to understand in order to progress, and I need to understand how to sharpen my teaching so no future student makes the same error. Let us together sort this out..."

Is there any danger that if the student graduates and propogates this misunderstanding, it could affect your reputation?

RBT said...

Yeah, just to support what Brad said. There are times, particularly in the middle of a long, tiring week of exams, when your brain can suddenly shoot off in a random direction on autopilot. Or you can read a question one (wrong) way, answer, and then forget about it because you think it's done and there are other, harder questions to focus on. I took a math standardized test many many years ago, and when I got back my answers I found that I'd gotten a "perfect"...except for a bunch of stupid answers in the "easy" section which I hadn't thought to check.

It certainly doesn't sound like anything was wrong with the question or the teaching, in this case!

a physicist said...

I can think of two more explanations. (1) Student was suffering from substance abuse issues. (2) Student was suffering from mental health issues.

I was on a faculty committee once which dealt with those two cases. I've known of a significant number of cases of (2), where students were having mental health issues and might take 1-2 semesters to get their meds balanced correctly. Not their fault, they had notes from doctors to back up this explanation. There were also other cases, too, where the student would explain "I decided to stop taking my meds which turned out to be a big mistake..." I think that these sorts of causes explain some of the strangest outliers in grade distributions and exam results.

MBF said...

I just finished teaching my first University course, a yearlong class for first-year students, and had a similar reaction to an answer on my exam. It wasn't just that his/her answer was wrong -- it was so spectacularly and confidently wrong that for a moment I seriously wondered if I was a terrible instructor who couldn't explain even the most basic concepts clearly. Fortunately the rest of this student's classmates got the question right or almost right.

In a way it's comforting to hear that more experienced professors go through this too.

Anonymous said...

As a PhD biochemistry major, question 1A was to write the dissociation constant. 1B was to write the association constant. To this day, I still can't figure out why I spent 1/2 of the exam trying to figure out 1B when I had 1A correct...

CSgrad said...

Another possibility: The student was ill (either acutely or chronically), and their physical discomfort was sufficiently distracting cause poor performance. Not necessarily ill enough that they would have thought to seek a postponed exam, but when you're tired and stressed even a bad cold, or normal flare-up of a chronic illness, can be distracting (been there, done that).

Hypatia's Ghost said...

Have you looked at all at misconception research in the educational research sphere? Sometimes, and in unexpected ways, people understand a concept completely wrong the first time, but it *sticks*. Then they spend lots of time and energy trying to fit everything else they learn in and around that misconception. If they never experience quite enough cognitive dissonance to make them revise the misconception, you can easily (and it's been documented to happen quite often!) have exactly what you've described happen.

A common one is the "seasons are caused by the Earth getting closer to the sun during the summer" thing. People can sometimes get confused when they're first learning about the Earth's orbit and axial tilt, and this seems like a reasonable statement to make. The important thing to remember is that they're not *dumb*, they just actually haven't had any meaningful experience with the concept that makes them address the illogic of their position.

Anonymous said...

"even professors who have been teaching for a long time may wonder (especially after grading final exams): Didn't I teach them anything? (Or the variant: Didn't they learn anything?)"

This rang so true. For me its the basics of genetics, that they "learn" first in high school, and which, for a senior biology major, should be written on their hearts, yet some of them still manage to erase the knowledge after EVERY exam.

Part of the reason is that we teach them too much, and they seem to have almost no ability to distinguish fundamentals and details. Another reason is that we teach them from third grade to learn for the test (thank you President Bush, among others). Finally, some of them simply don't care, but I am not so troubled by those folks.

However, we shouldn't let these answers make us forget the many excellent to outstanding students who really do care and think hard.

Mark P

Anonymous said...

Student could have had a preconceived notion of D that was never replaced. Was this concept tested earlier? Were there opportunities for you to discover that student thought D earlier in the semester. Preconceived notions only tend to go away when they are challenged in a conversation.

Lisa said...

How did the student do on the rest of the exam? Was it all as disastrous as this concept?

Anonymous said...

Similar experience to Brad. On an exam as a physics undergrad, a whole group of us turned three cubed into six... Luckily it wasn't just one of us and the professor found it more entertaining than anything else, and he chose to give us a lecture (while laughing at us).

As a faculty member now, I have seen similar things - and they have also been group effects on exams. Apparently simple math mistakes and exam brain freeze go hand in hand.

Anonymous said...

I could be wrong, but I would think that "math" errors (computation) are of a different sort than conceptual errors. ??

Anonymous said...

I bombed an exam as an undergrad - I really felt awful, but it was a midterm exam and I didn't feel like I could ask for an extension, so I figured I'd tough it out. I didn't say a word to the prof about this. Throughout the exam, I felt worse and worse. When I got to my dorm room I discovered I had a fever of 103; turns out I had a raging kidney infection... (Do you remember if this student missed this same question on every previous exam in these classes? I assume this concept has been tested before? If not, it may be that they were acutely ill, extremely upset for some reason, etc)

mathgirl said...

I agree with anon 11:23, the kind of math errors that are discussed (3^3=6) are cleary results of nervousness. I see them all the time in exams, and I usually deduct very few points for them (if any), unless the error makes the whole problem easier, but this does not tend to be the case.

The error that FSP discusses looks more like confusing a circle with a square...

John V said...

I'd guess the question and the answer are not as far apart and the misunderstanding not as great as you suspect.

Try asking the student his/her thought process. Perhaps the question was somewhat misread to ask something different, or the answer is a smaller perturbation on the truth than it appears.

I think, since we make up lots of questions knowing the answers, we think in our omniscience we can interpret students' thought processes better than we really can.

Although these answers gave me pause:
http://www.facebook.com/photo.php?fbid=10100278561457426&set=a.10100496694008046.2913504.2512625&type=1&theater

Barefoot Doctoral said...

As an undergrad, a moderate case of pneumonia that I was too stupid to see the health center about, caused me to fail a midterm in a class that I was otherwise doing particularly well in. I've also had brain freezes that have caused me to do the equivalent of 3^3-6 mistakes.

Having my own history of bad exams, on this side of the podium, I try to talk to students whose performances surprise me. Sometimes the answers I get are enlightening about my teaching. Sometimes they are just disappointing.

Female Science Professor said...

The student did OK overall on the exam. If there was a brain-freeze, it did not occur throughout the entire test-taking experience, and was only really spectacular on that one question.

Anonymous said...

I was asked on one exam for an explanation of nuclear fusion, and promptly proceeded to write a few paragraphs on nuclear fission. I aced the rest of the exam, though, so still got a decent grade.

Optixmom said...

Another issue I was confronted with was my blood sugar plummeted during an exam. I tried to hold it together long enough for it to pass. After that my brain was basically mush. The test was given in the morning from 9am until noon and the blood sugar thing happened at about 11am.

I have also experienced the brain freeze if I don't allow myself to recover from studying especially for an evening exam. Studying full throttle for hours and going right into an exam is not a productive method for me.

Stephanie said...

I too have messed up some fundamental and easy things on a test because it was too easy and I was expecting something harder. I tried to make it as hard as I thought the problem should be.

Anonymous said...

Don't over-analyze stupid answers unless you keep getting the same stupid answer from the same person over and over again (easier said than done, I know...).

I can verify from an unfortunate experience during my student days that it IS possible to easily answer all questions on a test correctly and have a complete brain meltdown on just one question.

This has only happened to me once, but to this day I can not understand what made me answer that one question as stupidly as I did. An answer that was so stupid that not only did it shock the teacher, who used it as an "how not to do" example in class(!), - it would make any normal adult, scientist or not, who has ever used a kitchen laugh hysterically!

gasstationwithoutpumps said...

@Mark P

Students do have a reason for forgetting everything they know right after the exam—it is a good strategy for the school game.

see
http://gasstationwithoutpumps.wordpress.com/2010/10/03/just-scoring-points/

EliRabett said...

aka head bangers

josie said...

I must admit I actually burst out laughing when I saw the diagram. It's also a relief that I'm not the only one who encounters things like this. I had a student who took gen ed with me, barely scraped by with a D-. Then, she went on to take a high stakes course (as in every aspiring med student needs) with me and of course, failed it. The strangest thing is, on the 3rd semester, she decided to retake the gen ed course with me (I assume to get a better grade). On the first exam regarding solubility and the concept of "like dissolves like", for a question asking which alcohol will be most soluble in water, she chose the one that's least soluble. At that point, I was truly puzzled. Even if she didn't pass the higher stakes course, that very basic concept is taught early on in both courses and it still did not sink it. To top it all, she failed the gen ed course she was retaking. Much as I would like to believe I have students who actually learn something from me, I also know there will always be those few who won't.

Bagelsan said...

I'd guess the question and the answer are not as far apart and the misunderstanding not as great as you suspect.

Try asking the student his/her thought process. Perhaps the question was somewhat misread to ask something different, or the answer is a smaller perturbation on the truth than it appears.


I like this idea! Personally I like short-answer questions (despite the threat of carpal tunnel) because there is a chance for the professor/grader to see the work; a lot of times a small mistake in an otherwise-decent thought process can result is some really embarrassingly wrong answers.

I've had situations where my understanding of a concept might be entirely perfect until the last bit, where I somehow get it into my head that "...and then you reverse it!" so I end up with something totally baffling if my procedure is a black box to observers.

EliRabett said...

One of the things that I do is to give multiple choice questions with a blank space underneath to show the work for partial credit.