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This fall I'll be teaching a standard "Introduction to Proofs" course and administration is advising us to be ready at any time to switch from in-person to fully online (or anything in between). I've been able to translate most of what I do into various modalities, but I'm having trouble with the memorization quizzes.

In a typical face-to-face version of the course I give weekly 10-question, 10-minute quizzes that require students to have memorized important definitions and/or theorem statements. I find this essential to developing their proof-writing skills. But if I give this in an online modality, what's to prevent them from just looking up the definition/theorem? (In my other courses I take the attitude that any online assessment is written to be open book/internet.)

Perhaps there's a better/alternate way to get students to really absorb the definitions/theorems and place importance on having them mentally readily-accessible. Perhaps in the modern age, all memorization is moot. I'm open to radical viewpoints/suggestions.

Note: I find myself rereading "How, now, shall we teach math online?" almost daily; mining for things I might have missed/forgotten and hoping that more people will post helpful ideas there.

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    $\begingroup$ You may want to abandon the idea of just seeing whether they remember the definition/statement, and instead ask them about whether they understand the basics of what it means. For example, if the definition is "injective", show the graph of $y=x^2$ and ask, "Is this an example or non-example? Explain your thoughts by appealing to the definition." $\endgroup$ – Brendan W. Sullivan Aug 7 at 17:38
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Here are some ideas:

  1. As Brendan says above, you can give an example or non-example, and ask the student to explain why (using the definition) it is an example or non-example.
  2. You can ask for examples satisfying certain properties ("Give an example of a relation on a three element set which is transitive and symmetric, but not reflexive").
  3. You can ask for critiques of fake student responses: "A student says that a number is prime if doesn't have any divisors. Is this a correct definition? If not, why not?"
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I've taught Linear Algebra a number of times. Teaching students to deal with proofs is an important part of that course. Luckily the textbook (Linear Algebra and its Applications 4th Edition, by David C. Lay) had true false questions in every section, and a bunch more at the end of each chapter. I used those to help students think carefully about the definitions and about proof.

I think you could come up with good questions whose answers don't appear online.

That will not prevent cheating, however. Students can get answers to newly posed questions pretty quickly through chegg and other cheating sites.

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Conducting tests in general in an online setting is hard, but especially checking things that the students should remember.
Some students will unavoidably cheat, but trying to minimize student frustration with the exercise can help to minimize it.
Having a longer exercise may seem to increase the chance of the student cheating but they actually make the exercise less stressful and by that minimizes cheating.
Having the questions be less about memorizing a 2 line definition and more about remembering the concept can help, by, for example, instead of “define a ‘transitive relation’” you could use “is the negation of a transitive relation transitive? Explain.”.

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    $\begingroup$ What is a transitive operation? What is the negation of an operation? $\endgroup$ – Steven Gubkin Aug 8 at 18:13
  • $\begingroup$ @StevenGubkin, Sorry for the wrong terminology, the negation is the relation is the relation who returns the negation of what the original relation returns. $\endgroup$ – razivo Aug 8 at 18:25
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You can't. It was a bad idea in the first place.

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    $\begingroup$ While I may agree with your first sentence, I would argue against your second. $\endgroup$ – Aeryk Aug 7 at 18:58

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