The prof I’m working with in our introductory #astro101 class at UBC surprised me today. I thought he was sabotaging a teachable moment when in fact, he pulled one of the most “agile” moves he’s made yet. Here’s the story:

Today is March 21, 2011, the first full day of Spring. The vernal equinox occurred yesterday, March 20 at 4:21 PDT. The instructor, let’s call him H, started today’s class with a clicker question:

The correct answer is A) but I fully expected a bunch of students to vote B), confusing the “going North” and “going South” for the Sun’s motion along the ecliptic.

The students thought, then voted. H looked at the results and said (I’m paraphrasing from memory),

The correct answer is A. 70% of you said that…

Oh, no, I thought to myself. He just gave away the answer and the success rate – only 70%, not terrific – and totally short-circuited the teachable moment that comes via peer instruction.

That thought took about 1 second, of course, so it was all over by the time H continued with

…Very few of you said B, C, or D and 30% said E. Let me show you one slide and then I’ll come back to the super moon.

You see, there was another event this past weekend. The full Moon occurred near perigee, the point in the Moon’s orbit around the Earth when it is closest. This means we had a full Moon, closer than usual, so it appeared bigger. Super, even. Oh, and it was.

So, here I was, getting alarmed that H was missing the opportunity for the students who voted A) to convince the students who voted B) to change their answers. But that’s not what happened at all. Hardly anyone voted B. They either knew the right answer A) or were more interested in the astronomy-in-real-life super Moon event. And H agilely, er, with great agility, confirmed the correct answer and followed up with an something 30% of the students cared about. He talked about the full Moon, how it was 14% bigger and 29% brighter. Not twice as big – don’t believe everything you hear on TV. That’s slightly bigger and closer than usual but not much. And no, the super Moon did not cause the earthquake in Japan.

Wow. I was impressed. He had the whole thing planned out but tailored his response based on theirs. Cool.

What about you? What teaching have you done, witnessed or experienced that shows agility?

How often have you heard your fellow instructors lament,

I don’t know why I bother with comments on the exams or even handing them back – students don’t go over their exams to see where they what they got right and wrong, they just look at the mark and move on.

If you often say or think this, you might want to ask yourself, What’s their motivation for going over the exam, besides “It will help me learn…”? But that’s the topic for another post.

In the introductory gen-ed astronomy class I’m working on, we gave a midterm exam last week. We dutifully marked it which was simple because the midterm exam was multiple-choice answered on Scantron cards. And calculated the average. And fixed the scoring on a couple of questions where the question stem was ambiguous (when you say, “summer in the southern hemisphere, do you mean June or do you mean when it gets hot?”). And we moved on.

Hey, wait a minute! Isn’t that just what the students do — check the mark and move on?

Since I have the data, every student’s answer to every question, via the Scantron and already in Excel, I decided to “go over the exam” to try to learn from it.

(Psst: I just finished wringing some graphs out of Excel and I wanted to start writing this post before I got distracted by, er, life so I haven’t done the analysis yet. I can’t wait to see what I write below!)

Besides the average (23.1/35 questions or 66%) and standard deviation (5.3/35 or 15%), I created a histogram of the students’ choices for each question. Here is a selection of questions which, as you’ll see further below, are widespread on the good-to-bad scale.

Question 9: You photograph a region of the night sky in March, in September, and again the following March. The two March photographs look the same but the September photo shows 3 stars in different locations. Of these three stars, the one whose position shifts the most must be

A) farthest away
B) closest
C) receding from Earth most rapidly
D) approaching Earth most rapidly
E) the brightest one

Question 16: What is the shape of the shadow of the Earth, as seen projected onto the Moon, during a lunar eclipse?

A) always a full circle
B) part of a circle
C) a straight line
D) an ellipse
E) a lunar eclipse does not involve the shadow of the Earth

Question 25: On the vernal equinox, compare the number of daytime hours in 3 cities, one at the north pole, one at 45 degrees north latitude and one at the equator.

A) 0, 12, 24
B) 12, 18, 24
C) 12, 12, 12
D) 0, 12, 18
E) 18, 18, 18

How much can you learn from these histograms? Quite a bit. Question 9 is too easy and we should use our precious time to better evaluate the students’ knowledge. The “straight line” choice on Question 16 should be replaced with a better distractor – no one “fell for” that one. I’m a bit alarmed that 5% of the students think that the Earth’s shadow has nothing to do with eclipses but then again, that’s only 1 in 20 (actually, 11 in 204 students – aren’t data great!) We’re used to seeing these histograms because in class, we have frequent think-pair-share episodes using i>clickers and use the students’ vote to decide how to proceed. If these were first-vote distributions in a clicker question, we wouldn’t do Question 9 again but we’d definitely get them to pair and share for Question 16 and maybe even Question 25. As I’ve written elsewhere, a 70% “success rate” can mean only about 60% of the students chose the correct answer for the right reasons.

I decided to turn it up a notch by following some advice I got from Ed Prather at the Center for Astronomy Education. He and his colleagues analyze multiple-choice questions using the point-biserial correlation coefficient. I’ll admit it – I’m not a statistics guru, so I had to look that one up. Wikipedia helped a bit, so did this article and Bardar et al. (2006). Normally, a correlation coefficient tells you how two variables are related. A favourite around Vancouver is the correlation between property crime and distance to the nearest Skytrain station (with all the correlation-causation arguments that go with it.) With point-biserial correlation, you can look for a relationship between students’ test scores and their success on a particular question (this is the “dichotomous variable” with only two values, 0 (wrong) and 1 (right).) It allows you to speculate on things like,

(for high correlation) “If they got this question, they probably did well on the entire exam.” In other words, that one question could be a litmus test for the entire test.

(for low correlation) “Anyone could have got this question right, regardless of whether they did well or poorly on the rest of the exam.” Maybe we should drop that question since it does nothing to discriminate or resolve the student’s level of understanding.

I cranked up my Excel worksheet to compute the coefficient, usually called ρ_{pb} or ρ_{pbis}:

where μ_{+} is the average test score for all students who got this particular questions correct, μ_{x} is the average test score for all students, σ_{x} is the standard deviation of all test scores, p is the fraction of students who got this question right and q=(1-p) is the fraction who got it wrong. You compute this coefficient for every question on the test. The key step in my Excel worksheet, after giving each student a 0 or 1 for each question they answered, was the AVERAGEIF function: for each question I computed

=AVERAGEIF(B$3:B$206,”=1″,$AL3:$AL206)

where, for example, Column B holds the 0 and 1 scores for Question 1 and Column AL holds the exam marks. This function takes the average of the exam scores only for those students (rows) who have got a “1” on Question 1. At last then, the point-biserial correlation coefficients for each of the 35 questions on the midterm, sorted from lowest to highest:

First of all, ooo shiney! I can’t stand the default graphics settings of Excel (and PowerPoint) but with some adjustments, you can produce a reasonable plot. Not that this in is perfect, but it’s not bad. Gotta work on the labels and a better way to represent the bands of “desirable”, “weak”, etc.

Back to going over the exam, how did the questions I included above fare? Question 9 has a weak, not desirable coefficient, just 0.21. That suggests anyone could get this question right (or equivalently, no could get this question right). It does nothing to discriminate or distinguish high-performing students from low-performing students. Question 16, with ρ_{pb} = 0.37 is in the desirable range – just hard enough to begin to separate the high- and low-performing students. Question 25 is one of the best on the exam, I think.

In case you’re wondering, Question 6 (with the second highest ρ_{pb} ) is a rather ugly calculation. It discriminated between high- and low-performing students but personally, I wouldn’t include it – doesn’t match the more conceptual learning goals IMHO.

I was pretty happy with this analysis (and my not-such-a-novice-anymore skills in Excel and statistics.) I should stopped there. But like a good scientist making sure every observation is consistent with the theory, I looked at Question 26, the one with the highest point-biserial correlation coefficient. I was shocked, alarmed even. The most discriminating question on the test was this?

Question 26: What is the phase of the Moon shown in this image?

A) waning crescent
B) waxing crescent
C) waning gibbous
D) waxing gibbous
E) third quarter

It’s waning gibbous, by the way, and 73% of the students knew it. That’s a lame, Bloom’s taxonomy Level 1, memorization question. Damn. To which my wise and mentoring colleague asked, “Well, what was the exam really testing, anyway?”

Alright, perhaps I didn’t get the result I wanted. But that’s not the point of science. Of this exercise. I definitely learned a lot by “going over the exam”, about validating questions, Excel, statistics and WordPress. And perhaps made it easier for the next person, shoulders of giants and all that…

I’m working with a veteran gen-ed astronomy (#astro101) instructor to make his classroom more learner-centered. We’re working hard on effective clicker implementation. The benefit of using clickers for think-pair-share (TPS) questions is the instructor can use the students’ votes to guide the instruction.

If everyone gets a question right, just confirm the answer and move on – don’t waste valuable class time re-teaching something everyone already knows! Conversely, if the students have no clue what the answer is and simply guess, you’d expect 20% for each choice A-E, 25% each if there are 4 choices, and so on. If that’s how they vote, either there’s something wrong with the question (a critical typo, perhaps) or the students haven’t learned the concept yet. Teach it again BUT NOT JUST LOUDER. Teach it again in a different way.

The “sweet spot” is when there’s a nice split between 2 or choices. The students have thought hard enough to formulate and pick the choice they feel is correct, which means they’re prepared to interact with their peers. In cases like this, we ask them to “turn to your neighbours and convince them you’re right.” Then you sit back and let them teach themselves. Ahhh.

(Well, actually, you shouldn’t sit back. You should wander around the room and eavesdrop – you’re going to hear some great ideas you can use for choices on the final exam!)

The hard part for instructors is knowing when to move on or when to get the students to discuss the question. Is 90% correct enough? Yes, probably. What about 80%? What about 60%?

In today’s astronomy class, the instructor asked the students a TPS question and the distribution of votes was A 0, B 0, C 67%, D 20%, E 13%. The instructor wasn’t overjoyed, but 67%? That means 2/3 of the students got it, right?

Wrong. Some knew the answer. And the rest guest. Er, guessed.

I did a little thought experiment with the instructor afterwards. “Suppose only half the students knew the answer and the rest just guessed. What vote distribution would you get?”

“Er, 50% then 10% for each choice, so a 60 and 10’s.”

“Great,” I said. “Suppose 2 of the 5 choices were obviously wrong. Then what.”

He thought for about 2 seconds. “67-17-17.” Our numbers from that today. “Oh.”

That’s right, when there are only 3 valid choice and only half the students know the answer, you still get about 67% success. And you might be tempted to move on even though half the students don’t know what you’re talking about!

That got me thinking – suppose fraction f of the students know the correct answer and the rest guess. What do the clicker vote distributions look like? I cast a spell with Excel (I’ve finally reached novice Excel spellcaster) and found these results:

(Quick limit test that us math-types do: when no one knows and f=0.0, the votes are 20% for each choice. And when everyone knows, it’s 100-0-0-0-0. Got it.)

For example, when the peak vote is 60%, only 50% of the students actually know the answer. And it gets worse when there are fewer choices (or equivalently, when you can eliminate some of the 5 choices because they’re obviously wrong.) Here are the distributions when there are 4 choice and 3 choices:

This last chart shows our 67-17-17 vote distribution corresponding to only 50% of the students knowing the right answer.

This isn’t ground-breaking research. I bet many clicker users have done this, too. Or at least, worked out a few special cases.

The moral of the story, though: the fraction of students who choose the correct answer is always higher than the fraction of students who know the correct answer. Don’t move on to the next topic unless you get a very strong peak.

What’s your threshold for moving on or doubling-back with a pair-share?