It wasn't until I wrote yesterday's blog post that I realized the extent to which I'm pushing inquiry-based learning in both courses I'm teaching this term. In both Precalc and Abstract Algebra I, the majority of the homework problems students are being asked to complete are what can legitimately be called research problems, and I'm posing them as such, guiding the students through an initial "data collection" stage, leading them then to a "conjecturing" stage, and from here to a point where they should be ready to offer at least a partial proof. The questions I'm asking are very open-ended, and in a few cases already this semester I'm not even sure I know the answer.
Example: I've got the Abstract students making conjectures about the relative primeness of consecutive terms in generalized Fibonacci sequences: for what natural-number pairs (α,β) is it the case that any two consecutive terms in the sequence defined by s0 = s1 = 1, sn = αsn-1 + βsn-2 are relatively prime? I admitted up front that I don't know the answer to this (though I have some guess as to what might be true), but I asked students to try out several cases, formulate a conjecture based on the data they gather, and try to prove their hypothesis.
What fun! I'm having fun, anyway. And what a way to learn! I have no doubt that the students are apt to become more talented mathematicians (and more generally, problem-solvers) when asked questions like this than when asked to complete cut-and-dried textbook proofs for which the answer is already told to them.
Thursday, September 15, 2011
Into the woods
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From my own experience working on the stated problem above, I have thoroughly enjoyed "researching" my given conjectures beyond the basic educated guess. It is one thing for me to guess that what I am conjecturing is correct, but another thing to prove that it is so. I am close to a full proof of my conjecture and I am overjoyed that I have been able to explore my capabilities to form such a proof! Thank you for the inventive homework problems, like this one, that challenge us.
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