Friday, February 27, 2009


R.B. Fuller conveyed a concept he referred to as "ephemeralization", which made the claim that as society improved its understanding of the world it, in fact, the efficiency with which it utilized scarce resources increased at an exponential rate, or certainly much faster compared to the rate of increase in the demand than is commonly supposed. In metals, for instance, recycling streams now make up a far larger percentage of production than in the past when ore was the major source.

Of course, as demand grows, supply must grow too or a shortage will ensue. But the relationship isn't a simple linear curve. I'm not sure if this is in any way original to Fuller, who made a career out acquiring ideas.

In any case, the effect of ephemeralization is in no way tied to a moral outcome. That is to say, there's no particular reason why one should think that changes in efficiency, in general, must be an increase which inheritly benefitting society. For instance, there has been a large-scale increase in enforcement activity of government over the decades, and masses of IRS laws passed which no one would pretend to understand, giving a bureacracy which adds nothing to the gross productivity of society yet which is more and more efficient at expending its capital. Another example is sometimes found in corporate quality organizations, particularly those involved with obtaining rubber-stamps of approvals for ISO9000 or CMMI, the motivations for which are anything but technical improvements to quality and performance. Most anyone who has worked in a large company will have had the experience of wondering just what the manager down the hall actually contributes to the bottom line, when he spends his days discussing the blueness of the sky or the methods of brewing sweet tea.

The point is, ephemeralization works both ways, simultaneously. Over time, many people will naturally find ways, in fact, to do less and less in exchange for the same amount of renumeration or more, and they are very creative in the manner in which they can fictionalize their actual levels of contribution. The more money available without true accountability, the less will be done with it. Oh yeah, that stimulus package is going to go far...

Saturday, February 21, 2009

Homework as Exercise

This is a third post on the subject, related to wasting time on homework exercises. Another way to look at this is by analogy to physical exercises. In general, one does not gain the benefits of a physical exercise without completing a full cycle of motion. When doing chin-ups, for instance, it isn't enough to just hang from a bar for two hours -- that's called stretching, and it does something but doesn't have the intended effect. One needs to completely move through a range of movement to get the muscle fibers to grow. Getting stumped on a homework problem and struggling with it for hours is like hanging from the bar... you end up sore with little to show for it.

On the other hand, muscle is gained by consistently overloading by a little more than your muscles are able to handle repeatedly. But you want to work around 70% of the one-repetition weight, so that you can still repeat the motions several times. Forcing yourself to bring to mind facts that you are trying to learn and string them together into very long complicated deductions is only helpful if the length of a deduction doesn't completely overload your working memory. When it does, your working memory fails, and you feel a sense of disorientation and dissatisfaction even if you get the deduction right. I would doubt you could easily commit such a deduction to long term memory either; the facts aren't all there in memory to reverberate together and establish a strong association to one another.

Learning as a puzzle; pedagogical misdirection and Rube Goldberg devices

Related to the previous post, I don't like puzzles. That is to say, it seems to me that as a pedagogical principle, making people find puzzle pieces that are completely unrelated to the concepts on which you are trying to focus is not an especially helpful practice. Perhaps this goes to a question of mathematical "maturity" -- and I won't claim to be an especially mature mathematician -- but using exercises which rely upon picking up on cues unrelated to the subject material seems to be deliberate misdirection.

The artist Rube Goldberg provided the quintessential characterization of modern pedagogical practice in algebra, calculus, and real analysis in general. While Goldberg's devices are often thought of as engineering constructs, they capture perfectly the actual deductive practices and proof methods taught in modern mathematical texts. The proofs we are doing in Real Analysis certainly get from point A to point Z, but quite often they do so only by introducing an almost comically complicated and laboriously contrived sequence of facts. Certainly, the lack of real comical effect may make it difficult for people to see the connection of proofs to Rube Goldberg devices, but they are both systems of logical deduction.

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,
(x12-2x1 +1) / x1 > 0
x1-2 +1/x1 > 0
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.