Saturday, February 21, 2009

Learning as an exercise in futility

It's been a relatively tough week, homework-wise. Not so much because of the volume of work, but because some of the assignments were puzzles with missing pieces. If had just been me, I'd not think much of it -- I get stuck on homework problems all the time -- but it was a shared experience. Many of the people in our Real Analysis class felt the same.

The first problem in particular stumped us for hours. It was a straightforward setup: show that a recursively defined sequence of real numbers is decreasing to a single number, and find the number. The sequence was defined as the first element being greater than 1 (x1>1), and subsequent numbers being 2 minus the reciprocal of the previous element (2-xn-1 ); we weren't supposed to use a closed formula (or calculus etc.) but deduce from a few limit theorems inductively. Now, the inductive step was straightforward, but we all got hammered on the base case: (x1>2-x2). There are probably many ways of doing this, but arguing on just an algebraic rearrangement of the inequalities proved to be a great way to whittle away hours of time. It felt like it needed a clever solution, and I just wasn't clever enough. After working out what I thought was a solution using a "dominate the terms" strategy, I put it away for the night feeling like I'd already invested too much of my time.

The next afternoon I began working on homework again, not having looked at the other problems yet... having wasted too much time on the first... only to discover that one term in my inequality was not right: I hadn't shown the base case. I got sucked back into it, thought I'd gotten it, then found a mistake, worked it more, found a mistake.... and then studied something else having discovered finally that I'd wasted another couple of hours. I regrouped with my homework team later, and the first thing out of their mouths was "I spent so much time on this problem, you wouldn't believe it!"

One consequence of continuing to work a problem after it has initially stumped you is that you are only learning to struggle and search, rather than learning the concepts which are the point of the exercise. There's one lesson here: when a problem isn't completely solved after a matter of minutes spent getting to understand the basic requirements of the setup, giving up early is a better strategy than struggling for hours.

Another consequence is that you are throwing out an unrecoverable resource: your study time. I have multiple classes, and time spent not being effective on one homework assignment is also time not spent at all on my other tasks. The same lesson is re-learned: give up on puzzles early, and use your time first for tasks you can complete.

One more observation here: If you cannot connect the dots from point A to point Z at a continuous pace during the homework assignment, you won't be able to do it later on a test anyway, so there's no use to completing an assignment this way. After completing other study tasks -- if you have time, return later and see if it is still a puzzle. If so, get help by talking with others. In our case, within about two minutes of meeting for our group homework, an algebraic relation for base case popped out for one of our team member's, and he scribbled the base case solution down. The relation was

x1-1 > 0

Well, this wasn't too revealing -- we all had mucked around with algebraic manipulation, but it happened that when we first met for the group session the second thing out our mouths was the strategy we were taking... we needed to show x1>x2, and so we needed either a direct identity or some bounding terms that dominated one but not the other... a > b > c means a > c and so forth... and that kicked the other student into recognizing that when squared so we also get that , (x1-1)2 > 0,
so
(x12-2x1 +1) / x1 > 0
so
x1-2 +1/x1 > 0
so
x1>2 -1/x1 =x2

Well, that seems pretty simple; just transform one relation into another by direct algebraic operations. The observation is that when you're dropping into one particular mode of solving a puzzle like this, it may be the wrong mode entirely, or at least one for which you might not have learned a large enough repertoire of the common patterns to be able to spot the necessary piece of the puzzle to stick in. Sometimes stepping back from the problem and discussing it is necessary to reset your mind set and re-think the approach, or just regain perspective.
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