Here is something I might try: show that one can fill up some region with disks of decreasing radii. This visual will demonstrate that the areas form a convergent series, but then we’ll show that the sum of the radii might well be divergent.
March 11, 2011
March 10, 2011
Students: do some problems without your book and notes!
I’ve found that some students make the mistake by always doing practice problems with their book and notes open. It is oh so easy to convince yourself that you know the material better than you actually do.
And yes, there is some actual evidence out there:
You don’t have to look far for instances of people lying to themselves. Whether it’s a drug-addled actor or an almost-toppled dictator, some people seem to have an endless capacity for rationalising what they did, no matter how questionable. We might imagine that these people really know that they’re deceiving themselves, and that their words are mere bravado. But Zoe Chance from Harvard Business School thinks otherwise.
Using experiments where people could cheat on a test, Chance has found that cheaters not only deceive themselves, but are largely oblivious to their own lies. Their ruse is so potent that they’ll continue to overestimate their abilities in the future, even if they suffer for it. Cheaters continue to prosper in their own heads, even if they fail in reality.
Chance asked 76 students to take a maths test, half of whom could see an answer key at the bottom of their sheets. Afterwards, they had to predict their scores on a second longer test. Even though they knew that they wouldn’t be able to see the answers this time round, they imagined higher scores for themselves (81%) if they had the answers on the first test than if they hadn’t (72%). They might have deliberately cheated, or they might have told themselves that they were only looking to “check” the answers they knew all along. Either way, they had fooled themselves into thinking that their strong performance reflected their own intellect, rather than the presence of the answers.
And they were wrong – when Chance asked her recruits to actually take the hypothetical second test, neither group outperformed the other. Those who had used the answers the first-time round were labouring under an inflated view of their abilities.
Chance also found that the students weren’t aware that they were deceiving themselves. She asked 36 fresh recruits to run through the same hypothetical scenario in their heads. Those who imagined having the answers predicted that they’d get a higher score, but not that they would also expect a better score in the second test. They knew that they would cheat the test, but not that they would cheat themselves.
Some people are more prone to this than others. Before the second test, Chance gave the students a questionnaire designed to measure their capacity for deceiving themselves. The “high self-deceivers” not only predicted that they would get better scores in the second test, but they were especially prone to “taking credit for their answers-aided performance”.
Bottom line: frequently quiz yourself by seeing if you can do problems without your notes or seeing if you can write out the proofs without references!
Note: the act of recalling the material actually has learning value too:
The research, published online Thursday in the journal Science, found that students who read a passage, then took a test asking them to recall what they had read, retained about 50 percent more of the information a week later than students who used two other methods.
One of those methods — repeatedly studying the material — is familiar to legions of students who cram before exams. The other — having students draw detailed diagrams documenting what they are learning — is prized by many teachers because it forces students to make connections among facts.
These other methods not only are popular, the researchers reported; they also seem to give students the illusion that they know material better than they do.
College Math Teaching 