“Another one of life’s disappointments.”
“What’s wrong?” I ask.
“Marking assignments, the bane of every teacher,” growls Ms. Spider, as she angrily scrawls the word “DREADFUL” on a sheet of paper. “This goose just divided by zero.”
I’ve always enjoyed math, but I am all too aware that it represents a bugaboo for many ordinary folk. Not everybody can have higher than average IQ and not everybody can play piano and solve Rubik’s Cube at the same time. I agree we have to Make Math Great Again.
“I s’pose I could improve my presentations skills or learn Statistics 101,” admits Ms. Spider.
“I confess I never studied stats at uni,” I respond. “I had to pick it up all by myself.”
“Learning stats 101 sounds too much like work. Surely there must be a better way.”
“You could make the exams and homework easier,” I suggest.
“We can’t make it too easy,” responds Ms. Spider. “I’m sure the good students wouldn’t mind an extra challenge or two,”
I steal a glance at the goose’s assignment. Yes the goose is below average, but one of the assignment questions are badly worded. Another question has kilometres as a typo for metres, and I have to suppress a chuckle. I can see why some of Ms. Spider’s students call her the WWMT.
“Actually,” says Ms. Spider, “I was toying with a more radical solution”
“We could give different exams to different students”
“What a revolutionary idea!” I exclaim. “Nobody has ever thought of this before!”
“From each according to his abilities … “
“From each according to his needs,” we chant in unison.
I am impressed: this Spider is clearly well-educated, not just in mathematics. She knows her clichés and sayings.
“Does that mean,” I ask, “if an awesome student correctly answers 40 assignment questions in a row then he will get a very difficult exam?”
“Hang on, what if an awesome student deliberately flunks the assignments …”
“Well … we could give the exam less weight than assignments,” the Spider responds somewhat nervously. “Then there is no advantage to tanking the assignments.”
“For this to work,” continues Ms. Spider, “we have to come up with some way of measuring the difficulty of certain questions.”
I mull over this for a while. We all know that students can be graded according to some chosen system. For instance, a math student can be Outstanding, Exceeds Expectations, Acceptable, Poor, Dreadful or Troll. But how can we grade certain questions?
The Spider writes two math questions on a sheet of paper:
“Which of these problems is harder?” asks Ms. Spider.
“I think both are equally easy. After all, I participated in the International Mathematical Olympiad many years ago.”
Somehow, I think that was not the answer Ms. Spider expected.
Behind us, a monkey, eagle, mouse, elephant, lion and jackal are enjoying some Texas Holdem. As usual, the monkey has squandered away all his chips early, and the Eagle is schooling the rest of the field, having accumulated more than half the chips in play. The Spider eyes them warily: clearly they should not be privy to our discussion.
“You see,” says Ms. Spider. “Sometimes I find it hard to judge the difficulty of a single question. For instance, I expect problem X to be easier than Y, but for some reason the reverse holds when I mark the assignments.”
I mull over Ms Spider’s words. I am not really in a position to judge, given I have never marked any student assignments.
“I have an idea,” says Ms. Spider. “Let’s draw a table”
“For simplicity,” says Ms. Spider. “Let’s assume each question is either marked correct or not correct, hence there are no partial marks. I use blank instead of 0 for ease of reading. Sam is an awesome student since she answered most questions correctly. Owen is a Stupid student because he only scored 2 out of 9. Each individual is represented by a single row.”
“But there is no reason we can’t do the same with columns if you pardon the double negative. For instance, only six people solved problem 8 but nine solved problem 9. Therefore problem 9 is harder than problem 8 …”
“So even if you don’t understand the questions themselves you can still say things like Debbie is better than Anna”
“Exactly,” replies Ms. Spider.
“With 18 students and 9 problems, you don’t have a lot of data”
It’s a stupid observation, I know – but I am only trying to buy time as I try to digest her ideas.
“Well, the same logic applies if we had 1800 students and 900 problems.”
“I think I understand,” I say. “It’s like some kind of Mechanical Turk. Previous students have tried these questions (and of course you don’t have to pay them to do these exams!), so you can work out which questions are easy or hard.”
“Wasn’t the Mechanical Turk some kind of fake chess-playing machine by Wolfgang von Kempelen? What a disgraceful idea! I would never try to cheat chess players like that”.
Okay, didn’t see that one coming. We need to agree on a definition of Mechanical Turk.
“Do you think your students will eventually find out their exam papers are different?”
“That shouldn’t be an issue,” says Ms. Spider, as she squirms in her seat. “If a poor student finds out, he has no reason to complain. If a good student finds out then deep down in his heart he already knows he is better than the poor student, so the exam result doesn’t matter.”
Somehow I think her logic is very, very, unsatisfactory. But I do know that many of the greatest insights precisely come from those who are willing to suggest ideas that sound utterly outrageous. For instance Rivest, Shamir and Adleman are your average computer scientists, but with a bit of luck they might one day become famous, known to every student of cryptography. So I should cut her some slack.
In fact, I am more than looking forward to the results of her revolutionary teaching methods. After all, I’m not the teacher and I don’t set the exams. I was especially careful not to suggest any drastic ideas of my own. If the radioactive 83,72,73,84 hits the fan and grows to fill the size of the entire house then I am more than happy to watch, knowing my 65,82,83,69 is fully covered.
Bring. It. On.