Question

When creating questions for coursework, exams, and similar, I sometimes want to task students with explaining counter-intuitive results or dissolving paradoxes, as I consider this is a good opportunity for them to train or showcase their understanding of the bigger picture or their skills to make an argument.

Now, a problem with just asking “Why is our intuition wrong here?” or “Solve this paradox.” is that it requires that everybody agrees upon what is intuitive or paradoxical here. A student may (quite rightfully) consider the correct result intuitive or just be happy with calculating the correct result and noticing that it is different from what intuition assumes.

Therefore I loathed such questions as a student since they strongly depend on interpretation. Can I somehow avoid this problem without dropping the respective task altogether?

Example: The Birthday Paradox

The birthday paradox is that n = 23 is the smallest number such that for a group of n people, it is more likely that two birthdays coincide than not. The number n is much lower than what most people would expect intuitively. I am seeking to ask a question that has an answer along the lines of:

While the probability that a given member’s birthday coincides with another only grows slowly (linearly) when the group size is increased, birthdays can also coincide between newly added group members. The number of such pairings grows faster (quadratically).

Now if I ask the students: “Why is n so low?”, they could ask: “In relation to what?”. Also, they might ‘simply’ calculate n, and I could not rightfully complain that this is not what the question asked for. (Of course, correctly calculating n in this case requires including pairings between newly added nodes and hopefully grants the students the kind of insight I am aiming at. However, some may still apply some formulas blindly and this is not so obvious in more complicated examples.)

  • 1
    Since such paradoxes occur primarily in logic and math, isn't this better asked at mathematics.se or matheducators.se? There are other sorts of faulty intuition, of course. – Buffy Aug 15 at 20:20
  • 2
    @Buffy: Paradoxes or counter-intuitive results exist in many disciplines. They may be more likely to be called paradoxes in mathematics, while in other disciplines it’s just considered an counter-intuitive result. As a matter of fact, this problem arose when teaching physics. – Wrzlprmft Aug 15 at 20:42
  • I don't really understand the purpose of the question. The birthday problem is a classic in learning probability and teaches the lesson that computing the probability that something isn't true is equivalent to finding the probability that it is. But once that lesson is learned it is a powerful tool. I don't see how you intend to generalize that very specific insight to get to a general question here. – Buffy Aug 15 at 21:08
  • @Buffy: I fail to see how the first part of your comment implies that my question is not generalisable. In fact, the birthday problem is not amongst the actual problems I am dealing with; I just chose it as an example because it is comparably low-level, well known, and easy to grasp. – Wrzlprmft Aug 15 at 21:44

Let your students debunk a wrong argument

Present a specific faulty argument based on intuition. Then let your students find the flaw in it. This way you relieve your students from the burden to divine what you consider intuitive. They also have a more specific task that cannot be solved by just deriving the correct result.

For example a task for the birthday paradox could be:

Bob argues:

A random person has a different birthday from me with a probability of 364/365 (ignoring leap years, seasonal variations, etc.). Hence, if I am in a group with n−1 other people, the probability that all of them have a different birthday is (364/365)n−1. This probability is first smaller than 0.5 for n = 254.

Find the flaw in Bob’s argument.

  • 4
    I doubt that this helps them learn. In particular, by your framing of the problem you are leading them to think in an incorrect direction. That makes the problem harder. That is quite different from asking them to debug an already incorrect intuition. Moreover, if you do this sort of thing, you need to have some assurance that correctly finding the flaw will let them make the correct inference rather than just being lost. Of course, if is "just for fun" then there isn't any issue. Caution advised. – Buffy Aug 15 at 20:49
  • Depending on the case, finding the correct result may be another exercise, has already happened, or is very tedious and unenlightening. It’s not that I would put such a task in a vacuum. – Wrzlprmft Aug 15 at 21:01

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