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Hypothesis behind “Pangs, Pings and Pongs”

True or False

All pangs are pings.
Some pings are pongs.
Therefore some pangs are pongs.

In Pangs, Pings and Pongs, I mentioned that it was motivated by a hypothesis about how people would approach solving such a problem. This is communicated by Gary Antonick of the New York Times NumberPlay blog.

There are three main ways to solve this problem:

  • Apply a memorized rule: All P are Q, etc (Raj Magesh, Agnishom Chattopadhyay, Vedanth Bhatnagar, ...)
  • Create a venn diagram (Gautam Sharma, Chew-Seong Cheong, Gregg Iverson, ...)
  • Use an analogy (Amelia Liu, Ian Mckay, Trevor Arashiro, ...)

The questions that he wanted to study are:
1. What approaches are used by high-school students? Undergrads? Research faculty?
2. Does country (education system) make a difference? If so, why?

The hypotheses are:
1. Research faculty will move between analogies and diagrams, using both.
2. Students from certain education backgrounds will avoid the blind use of formulas.
3. Students from certain education backgrounds will be more likely to relate it to what they have learnt, and thus rely more on analogies.

What are your thoughts?

Note by Calvin Lin
2 years, 5 months ago

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I think this is an interesting study to conduct, because it could potentially highlight shortcomings in math education. When I read that problem, I immediately first thought of an analogy, but just to be sure, I quickly sketched out a Venn diagram. There's almost no problem in logic that can't be tackled by use of Venn diagrams. Meanwhile, there's predicate logic formalism, which is great if you want to axiomatize this kind of thing, with the objective of mechanizing it for far more complex problems in logic that would be unwieldy to do using Venn diagram sketches. The point is that we really should be familiar with all the approaches, and more still if we could. We do want to emphasize the fluidity of taking different approaches, because that's what mathematics is all about. Why do we have students looking at problems like this only in a certain way but not others? I wonder if schools are failing to explain that there can be a variety of approaches in solving a problem, because it makes it difficult to "test and fairly grade" students. To use an analogy, if a stopwatch is used to measure all athletic performance, how do we measure dance?

Edit: Here's an example of going off on another approach to working with logic problems:

Propositional Logic Using Algebra

Michael Mendrin - 2 years, 5 months ago

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There is some principle to solve this problem if you know please elaborate it

Kutumbaka Jaswanth - 2 years, 5 months ago

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i used kinda analogy

Abu Zubair - 2 years, 5 months ago

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I always use diagrams while solving such riddles. It becomes really easy then!

Swapnil Das - 2 years, 5 months ago

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I used Analogy. BTW is there a rule for such type of questions??If yes,Can some, please elaborate it.

Kalash Verma - 2 years, 5 months ago

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