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1/n I believe very strongly in learning guessing. But how do we teach this in the classroom?
Here’s a small example, which happened as part of a precalculus review for calc I students a couple of weeks ago:
Here’s a small example, which happened as part of a precalculus review for calc I students a couple of weeks ago:
2/n The first class, I solicited guesses for the answer to the following:
find two numbers whose sum is 5 and whose product is a maximum. 2 and 3 was suggested, followed by the question: do they have to be integers?
This led to a guess of 2.5 and 2.5.
Next question and guess:
find two numbers whose sum is 5 and whose product is a maximum. 2 and 3 was suggested, followed by the question: do they have to be integers?
This led to a guess of 2.5 and 2.5.
Next question and guess:
3/n Given 600 feet of rope, form a rectangular swimming area using the beach as one side of the rectangle. Find the largest swimming area.
The guess of a 200 by 200 rectangle was provided and found incorrect, using precalculs methods.
Why was this an incorrect guess?
The guess of a 200 by 200 rectangle was provided and found incorrect, using precalculs methods.
Why was this an incorrect guess?
4/n They figured it out; the beach provides “free rope”, making this question fundamentally different from the original question about the product of numbers summing to 5. (The 1st Q. inspired guessing a square.)
Things got better when we revisited this during the next class:
Things got better when we revisited this during the next class:
5/n Since we had not used formal methods on the first example (ie we had not yet proven that the product was maximized via 2.5 and 2.5) we revisited it more formally, but then I threw in an observation: 
6/n (The observation was in regards to the beach example). But then this led to an interesting guess about the first example:
7/n Since writing the function as a constant (6.25) subtract a square (necessarily at least 0) guarantees that 6.25 is the maximum, it actually happened in class, that *GUESSING* the answer led to a *PROOF* that the guess was correct.
8/9 So this wasn’t a refined comprehensive method for studying quadratic functions, but *it was fun*.
It also emphasized the attitude that if you use everything at your disposal—including memories, hunches and guesses—you can find an interesting truth by making connections.
It also emphasized the attitude that if you use everything at your disposal—including memories, hunches and guesses—you can find an interesting truth by making connections.
9/9 And yes, I’ve certainly used this approach in research as well. To the extent that knowing more helps one prove things, guessing can be a great substitute until a proof is within reach.
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