Reading Log: Outliers

I have several rather large stacks - well, more like an amorphous pile at this point - of books that I've finished reading but haven't written about. I had intended this blog to act as a reading log, among other things. Not writing about books that I finish doesn't contribute much to that goal.

As an insignificant token gesture toward rectifying that, here's a book that I borrowed, and which I'd like to return: Outliers: The True Story of Success by Malcom Gladwell.

This one is borrowed from the local library. I've already kept it past its due date, and renewed it, and am fast approaching its new due date. I had read it well before the first due date rolled around, but held onto it in faint hopes of writing this entry.

There's a library very close to my workplace. So close, in fact, that I can spend my lunch breaks there, and most days I do. I have a set prioritization sequence. Each day I read the newspapers (including any from past days that I haven't read yet), then any new periodicals of any interest to me whatsoever, then scan the new arrivals shelves for anything that looks interesting, then finally work my way down a shelf, in Dewey Decimal System order, at least skimming each book. I keep notes on that last method, so I can remember the shelf and book where I last left off; most days, I don't make it past the new arrivals, so I rarely resort to this method.

I'm usually content to read a new book for a few minutes to get the idea, maybe returning to it another day if it's still there. Outliers has been the only book so far for which my ten-minute skim wasn't enough. I signed it out so that I could finish it at my leisure (which took two or three days).

This books starts strong, with a chapter on a phenomenon that fascinated me: the tendency for elite athletes to have been born in the early months of the year. The author provides a few lists of professional sports team rosters, and it's immediately obvious that the players are far likelier to have been born between January and March than at any other time. Then he explains why.

It's one of those things that's so obvious, you wonder why we hadn't seen it all along: at first, in any group activity, children are grouped by age, and there has to be a cutoff. Whether we're talking about junior sports teams, school, or almost anything else, all the kids born before an arbitrary date go into group A, and all the ones born on that date or later go into group B or wait until next year. It quickly becomes apparent that some of the kids in each group are more talented than others. Those more talented kids get more attention, more opportunities, better coaching, more practice time, whatever. It's only normal to encourage and nurture talent. The less proficient kids fall by the wayside. They may get relegated to the backup team, put into a less challenging curriculum, held back for remedial work, etc.

However the truth is that those "more talented" kids may not be prodigies at all. They may just be older.

These groupings by birthdate usually happen at an early age. Five or six years old, at the latest. When grouping those children into age cohorts, you end up with some children in the group being as much as one day short of a full year older than the others. At age five or six, a gap of (effectively) a full year makes a huge difference. The child who is five years, 11 months, and 28 days old will be bigger, stronger, faster, more co-ordinated, more agile, and more intelligent than the child who is five years and 2 days old. They will be "better", by almost any measure, by a significant margin.

That younger child will wind up on the losing end of almost any comparison. Before long, they will be shunted off into the second (loser) division. The "prodigy", whose true main advantage was that they had several extra months of development under their figurative and literal belt, will receive better training, more resources, more opportunities, and ultimately a better outcome.

This is not to say that birthdate is the only predictor of success, that children born at the "right" time never have true natural ability, or that children born at the "wrong" time can't sometimes overcome that through true natural ability or sheer determination. However, it's clearly an important factor.

The only hockey player whose name I could immediately think of when I first read this was Wayne Gretzky. He was born on January 26.

I found this age issue mind-blowing, and I wonder how it could be overcome or, to put it more crassly, perhaps even exploited. The first thing that comes to mind is that a rival children's sports league could be started, with the age cutoff exactly six months off from that used by the current, established league. If you could get this to take hold (and you'd have to follow it up through all the age groups for such a league), I think that within a few years we would discover almost twice as many "prodigies" in the sport. The ones born in January or soon thereafter, and the ones born in July or soon thereafter. How many potential Wayne Gretzkys have gone unnoticed simply because they were a few months less developed?

That's in sports, though, about which I give not the slightest whit. It's just an area where this phenomenon is easily observed. Gladwell - among other sources we'll discuss shortly - assures the reader that statisticians have done proper studies on the matter that demonstrate a clear, strong correlation between high athletic achievement and a birthdate near the entry point cutoff.

I'd be far more interested in overcoming this in the academic world. Many a child has been pigeonholed early on by the bureaucrats in charge of the school system, and proceeded to live down to the expectations placed (or not) upon them. Just one more way in which the public "educational" system fails miserably.

I'm something of a numbers and statistics geek, so when I read about this in Gladwell's book, having never heard of it before, I thought it must be a very obscure phenomenon. Then, shortly thereafter, I read about it on Cracked.com .

I think it's safe to assume that if a research area is being discussed on Cracked (which is sometimes a surprisingly good source of information), it's not exactly a secret anymore. I was just late to the party.

The rest of Gladwell's book is still somewhat interesting, but didn't grip me like that first revelation. He debunks the myth of effortless expertise, by using such examples of Bill Gates and the Beatles. Gates was certainly brilliant, but to suggest that he was an inexperienced natural who mastered computers by sheer intuition is off the mark. He had several opportunities fall into his lap by various means that allowed him to rack up thousands of hours of programming experience at an early age. Where most of us squander our time, young Gates passionately threw himself into a hobby that turned out to have a very practical application later in his life.

As for the Beatles, they honed their craft with months of nights on Hamburg nightclub stages, playing for eight to ten hours a night, seven nights a week, for demanding audiences. They didn't get as good as they became without paying years of dues. By the time they appeared on the Ed Sullivan show, they had spent more hours onstage than most veteran performers will amass in their entire career.

The books bogs somewhat in the middle, as Gladwell spends far too many pages explaining why Jews became so prominent in the legal profession (no, really, he does). The book is well worth reading overall, but I could understand a reader setting it aside after the chapters on birthdates and "overnight success stories".

Another fascinating section comes when he compares two brilliant physicists, Chris Langan and Robert Oppenheimer. Oppenheimer was somewhat deranged, even trying at one point to poison an associate for no comprehensible reason, but was put in charge of the Manhattan Project. Langan falls victim to a series of seemingly minor misfortunes, and winds up a footnote. The main differences between them lie in their social skills - Oppenheimer is a master manipulator, although we, as always, pretty this up by calling him "charming", "personable", or "persuasive". As such, his flaws, though at least comparable to those of Langan, are overlooked whereas Langan is allowed to drift into obscurity.

Much of life is a popularity contest.

I also appreciated the theme that raw intelligence ultimately counts for little. There seems to be a point where IQ is simply "high enough", and being any more intelligent past that threshold is insignificant. There is also a large degree to which, as in the case of Oppenheimer and Langan, raw intelligence matters less than "social skills", a euphemism for likability and capability to manipulate.

I appreciated this because I'm a guy whose IQ consistently tests in the top 1/2 of 1% of the population. It's been formally tested a few times over the years, and I've taken a few other informal but supposedly valid tests as well, and every time I've scored in that range. In real life, that counts for the exact midpoint of jack and squat.

I have a nice, nondistinguished, middle-class existence, which suits me fine. However, it's certainly not what anyone, including myself, would have predicted for me back when my elementary school tried to move me up several grades (Mom vetoed it), but settled for letting me take individual advanced classes (grade 5 English when I was in grade 1, high-school English and math in grade 3), or when universities began calling my house when I was still in junior high school. As it turned out, my academic career was solid but nothing special, and my professional life has been similar. My social skills are closer to those of Langan than Oppenheimer, so it could have been much worse, and I have absolutely no complaints about my lot in life. Still, one of my favourite TV shows was Malcolm in the Middle, because of its recurring theme: being intelligent doesn't make you smart.

Back to the book. The question I'm left with is what we can do with this information. By the end of the book we know that grouping children by age conveys a huge advantage on the oldest members of the group. We know that it takes approximately 10,000 hours of experience in anything to become an expert, and there are no shortcuts. We know that sometimes the skills acquired to adapt to adverse circumstances pay off handsomely later, when the circumstances change (the point of the section about Jewish lawyers).

So, do we stop grouping children by age? Do we devote more time to productive or educational activities, which sounds more than a bit obvious, no matter how unlikely? Do we focus more on developing our interpersonal skills? Do we contact our friendly neighbourhood psychics to ask what things will be like in twenty years, so we can start angling into position now?

The only practical action I can see is to try to provide opportunities to others. Part of Gladwell's thesis is that his titular "outliers" may not be so special in and of themselves. They may have been in the right place at the right time or willing to do what needed to be done to advance. However, many of them reached in their positions of prominence due in no small part to the largesse of others. No one really makes it alone. Bill Gates was given time in computer labs in an era when that was a rarity. A club owner decided to give the Beatles, a moderately talented garage band like hundreds of others, a shot as his house band. A blind eye was turned toward Oppenheimer's antisocial tendencies.

Help one another. Give a break to somebody who could use one.

I guess that's as good a moral to the story as any.


Enough rambling. Here's a picture of my new dog, back when she really was my new dog, in every sense of the word. She's a little over a year old now, and her head alone is now much larger than those boots on the left.

Yet Another Triumphant Return

I recently returned from a couple of weeks of work-related travel. I had a couple of noteworthy experiences on the trip (besides the stuff that I won't be discussing on here for various reasons).

I almost got taken down by airport security at my point of departure. Besides my checked suitcase, I had a carry-on bag. It was a backpack containing my Bible and portable DVD player, and a few other books to occupy me during the trip.

The basket containing my jacket and the stuff out of my pockets passed through the x-ray screening without difficulty, and my own passage through the metal detector was also uneventful. The backpack, however, proved to be a problem. I had to put it on the little conveyor belt, and when I went to meet it on the other side a security guard was waiting for me, already holding it.

"Do you have a multi-tool in here?" he asked, in a pleasant enough tone.

"Nope," I replied confidently.

He smiled, not unkindly. "Are you sure?"

"I was, but I'm getting less sure all the time."

He chuckled, and began unzipping compartments. "Looks like it's way down at the bottom somewhere." After a minute of digging, he produced a multi-tool in its carrying case. It was the one I carried on my belt during my IT days. When I left that job (a diplomatic way of putting it), I stuck the multi-tool into the backpack, which I haul around with me anytime I go anywhere with stuff to carry. I figured that way the multi-tool would always be handy if I was out somewhere and needed it. Of course, I forgot all about it being in there when I was packing for this flight.

"I can assure you that I wasn't trying to smuggle that onto the plane. Feel free to chuck it, or whatever you do with seized contraband."

The security guard was very nice about the whole matter. Instead of just taking it, he asked if somebody had dropped me off at the airport, and if so, whether they were still around. My wife and mother-in-law were just outside the security screening area. I pointed them out, and the guard had someone take it to them. It was waiting safely for me at home when I returned.

Good thing I speak English, or I probably would have died right there, twitching on the airport floor.


Even weirder, the same bag got stopped again by security at the airport for my flight home. Once again I put it dutifully on the conveyor belt. This time, the young lady watching the x-ray monitor (which I couldn't see) stopped the belt and looked at the monitor for a while, clearly puzzled. She was tipping her head to the side, reminding me of my dog's reaction when I used to take my video camera, record myself calling her, and play it back on the living room TV. "Wait... you're on the TV, calling me, but you're sitting over there... but you're on TV... but... now my head hurts. Good thing my walnut-sized brain means I'll forget this in ten seconds, or I'd be traumatized."

The screener lady called a colleague over to join her for some synchronized head-cocking. Eventually they decided to send my backpack to someone else. It came out of the x-ray machine, and was promptly grabbed by a very serious looking guy, who said, "We've got to test this."

"OK", I cheerfully replied. I had lots of time before my flight.

This new guy dug through the bag, and ran a little wand over it. I don't think it was a metal detector. My theory is that it was a dowsing rod, and he suspected that my backpack contained an underground spring. In any case, after a few brow-furrowed moments, he handed me the backpack and said, "OK, you can go." This guy was just gruff enough that I decided not to push my luck by asking any questions. The folks at the first airport, which was much smaller, were a lot friendlier, and they busted me trying to sneak a weapon onto the plane.

It looks like I'll be flying again in October. Perhaps I should invest in some less suspicious carry-on luggage before then.


Enough rambling. Here's a picture of the bottom shelf, left-hand side, on bookshelf # 1. On the left are a bunch of records that don't fit on my actual records rack. Most readers under 40 will have no idea what "records" are. In the middle are notebooks and photocopies of textbooks from my university days. Yes, photocopies. I was blatantly disregarding copyrights long before anyone ever heard of Napster. Photocopies were a dime each at the library photocopiers (I liked the unsupervised one in the basement), and reduction allowed two-page spreads to fit onto a single sheet while remaining legible. Even those of us who weren't math majors got pretty good at calculating whether it was cheaper to buy a textbook or just photocopy it. Finally, that pile on the right is what remains of my Rolling Stone magazine collection. I got rid of the vast majority of them long ago.

Reading Log - Mark Haddon's Clash Of Fundamentalisms

I'm back. I may talk about the trip a little some other time. For now it's Sunday, and I've got a reading log entry with potential for spiritual implications, so away we go.

One of the books I read during my exile was The Curious Incident of the Dog in the Night-Time, by Mark Haddon. It's the story of an unusual young man's investigation into the "murder" of his neighbour's poodle. The young man's name is Christopher, and he's autistic or has Asperger syndrome. The book is written in first person from Christopher's perspective, and he has no interest in any labels anyone else may place on him, so his exact diagnosis is never revealed.

From here in there may be spoilers for anyone who intends to read this book but hasn't yet. If you happen to be married to me, this means you.

The book is compelling and Christopher, although a bit odd by western social standards, is a well-drawn character. He is extremely literal-minded and uninterested in emotions. The other characters in the book are not fleshed out nearly as well, but this too rings true since everything is narrated by Christopher, who isn't particularly interested in or capable of understanding what's happening inside anyone else's head.

Christopher will ring true to anyone who is close to someone on the autism spectrum (or on it themselves). He is single-minded in his quest to determine who killed the dog, finding loopholes worthy of a high-priced attorney in his father's orders to leave the matter alone (and thereby ceasing to pester the neighbours with questions about it).

About halfway through the book, there is a sudden shift. It's not a Shymalan-level twist, but something is revealed that simultaneously makes perfect sense and alters the course of the narrative. The book is no longer about what it had been about. It gets much better, although it had been perfectly good to that point.

I won't reveal the twist, or even whether Christopher solves the crime. Instead I want to shift gears, using the book as a segue to my justification for posting this on a Sunday, the day I set aside for spiritual themes on this blog. Part of the book's structure is that the chapters (which are numbered in ascending prime numbers, one of Christopher's obsessions) alternate between moving the narrative forward and Christopher explaining his take on some subject, often mathematical.

In several of the "explanation" chapters, Christopher mentions that he is an atheist, and goes into more detail in a few. He rejects the Bible as having any divine origin, dismisses any talk of Jesus as fable, and accepts the theories of the Big Bang and evolution as perfect, comprehensive explanations for the origins of everything.

In our modern, "enlightened" world he is far from alone in these positions. Many people agree with his articles of faith. They cling to Science as fervently as any other fundamentalist clings to their dogma. They consider themselves different from, and even superior to, those other fundamentalists because their beliefs are based on Science, not on the unverifiable writings in an ancient tome.

Their logical error here is obvious. When it comes to matters like the origins of physical reality or the human species, the writings in any science book are speculative. The theories are no more testable, no more provable, than Genesis. Whenever a physicist begins expounding on how the universe was created, utterly certain, utterly unshakable in their faith, I'm always reminded of Job 38:1-4:

Then the LORD answered Job out of the storm. He said: "Who is this that darkens my counsel with words without knowledge? Brace yourself like a man; I will question you, and you shall answer me. Where were you when I laid the earth's foundation? Tell me, if you understand."

No, I don't have all the answers to every question in life. I can't ultimately be 100% sure that my beliefs are correct. Neither can anyone else. The primary difference between a dogmatic materialist and any other believer is their choice of sacred text. I've chosen the Bible; others choose the writings of Darwin and their high school textbooks. I note that those schoolbooks invariably get revised. I've read several science books that had forewords explicitly stating that most of the book would eventually be supplanted by newer, more correct information. I've never seen a Bible with that sort of disclaimer, and don't expect to ever see a new edition with corrections. The Bible doesn't need it.

Here's another key difference between a hardcore materialist and other breeds of fundamentalists: the materialists tend to be much more certain that they're right, and much more smug and arrogant about the discrepancies in beliefs. If I didn't believe in the depravity of humankind, I'd probably be confused when more "enlightened" people almost invariably prove themselves less tolerant than supposedly backward believers.


Enough rambling. Here's a picture of the fourth shelf down, left side, on bookshelf # 1. I don't seem to have a picture of the third shelf, so we'll come back to it another time. This shelf is full of psychology and criminology textbooks. The black binders on the right are full of Sunday School materials that I've written over the years. Those binders contain a lot of plastic sheets for overhead transparencies, which gives an idea how old they are. Nowadays I usually don't even bother keeping a paper copy of the handouts, and just keep the OpenOffice files.

Marital Mathematics

It's very close to my wedding anniversary. My wife’s, too. What are the odds of that? This anniversary is a round number, one of the Big Ones. My wife likes to call it “Ten down, two to go!”

The actual anniversary went by very recently. Or it might be coming up very soon. I forget. It might even be today, which would plausibly explain all the people in my house and possibly the balloons. However, I still haven’t gotten a satisfactory explanation for the streamers. I’m pretty sure they’re unrelated, and reject all claims to the contrary.

This is the kind of occasion when one reviews and takes stock of their life. In my case, it makes me wonder about a specific statistic. This is a delicate matter, but it should be clear from this blog that any aspect of my life, no matter how intimate, is fair game for discussion. Well, except for anything that’s actually personal.

Anyway, I got to wondering about, shall we say, “relations” statistics for my wife and I over the course of our marital career (sorry for the vulgarity, Mom!). If we appeared on baseball cards, this is the kind of thing I’d like to see on the back. So, I ran some numbers to estimate how many times we’ve (THIS PORTION OF THIS BLOG HAS BEEN REDACTED BY THE DEPARTMENT OF MORAL STANDARDS ON GROUNDS OF GENERAL ICKINESS) over the last decade.

Ten years is 520 weeks, or about 3,640 nights. So, divide that out, carry the eight…

Four times. Yep, that sounds about right.

No, wait, three. I forgot that 2000 wasn’t a leap year.

In other news, reliable reports indicate that Michael Jackson remains dead. However, several witnesses claim to have seen Generalissimo Francisco Franco at the 7-11 in Des Moines, Iowa on Monday. There was a special on Slush Puppies.

We have received no new information on the whereabouts or condition of Hostage Bunny. Our prayers tonight are with Mrs. Bunny, their 74 children, and their innumerable grandchildren.

Enough rambling. Here's a picture of a finished frog pinata. Sadly, no photos were taken of the smashed-to-bits frog pinata that came to be soon thereafter. There was green paper mache and months-old candy (mostly hard candy wrapped in cellophane, so no worse for wear) everywhere.

Bible Defense: Everybody Likes Pi

Today's Bible Defense article was inspired by a comment on my explanation of the birds and the bats. In the comments on that article, someone with the handle Evil Bender posted this:

Of course, one could ask why God would have the Hebrews using a word that groups animals by flight, not by relationship. Surely he could have given them a better understanding of nature?

That the Creator of the Universe would conflate bats with birds (while apparently excluding flightless birds) does not help your case for infallibility.

Nor does the fact that pi is not equal to 3, nor that the gospels disagree about the event leading up to Jesus' death.

And that's without even getting into different lineages, iron chariots, and heaps of other biblical contradictions.

First of all, I'd like to thank E.B. for making a polite, thoughtful contribution to the discussion. All too often this sort of thing turns into a flamewar, consisting of nothing more than name-calling substituting for argument. E.B. has not done that, and I certainly don't intend to either.

Before we get into the meat of this new entry, I'd like to reiterate the point I wanted to make about bats and birds: there was no need for God to explain that bats aren't birds, since the ancient Hebrew word translated as "birds", 'owph, does not mean "birds" in the modern sense. It would have been completely nonsensical for the Bible to stop at that point to explain that bats are not birds, since in the original language "birds", in the sense of the category as it now exists, was never used. If the word "birds" is a problem, then forget that it's there, and mentally insert the word 'owph at that point.

In short, "birds" is an imperfect translation of the original word, which in no way meant or implied that only feathered egg-laying animals were intended. There's no need to defend the idea that the Bible calls bats "birds", because it doesn't. It calls them 'owph, which they are, and we translated it imperfectly.

Put another way: "That the Creator of the Universe would conflate bats with birds..." He didn't. Those who don't understand that 'owph does not mean "bird" did.

I don't know how to put it any more clearly.

However, E.B. raised another point that I'd like to address: the value of pi. His other alleged discrepancies fall into the category of "stuff I haven't researched or thought about enough to discuss intelligently". Maybe some other time.

Many people over the years have argued that the Bible says that pi is equal to three, based on this passage, I Kings 7:23 (repeated in II Chron. 4:2):

He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it.

Before we get into why this passage does not say that pi is equal to three (and why some people think it does, in case you've put all that stuff from geometry class behind you), let's talk about some of what those things mean.

Ritual washing was a huge issue for the priests of ancient Israel. There are many passages, especially in Leviticus, that go into great detail explaining exactly when and how priests are to wash before carrying out their duties. Most of the explanations boil down to cleanliness being a symbol of holiness; it would be an insult to God to appear for His service looking anything less than one's best. The phrase "cleanliness is next to Godliness" does not appear in the Bible, but it's easy to see how people make that logical leap.

The principle still holds today. Nobody wants to show up for a job interview, or the first meeting with the parents of someone they're dating, with a stain on their shirt or a milk mustache. It would be embarrassing to appear before someone whose approval you seek without cleaning yourself up.

The Old Testament rules of washing served a health purpose, as well. The priests handled food that was offered as sacrifice - which was then eaten. Long before humans understood germ theory and bacteria, God was promoting safe food handling techniques.

"The Sea" was a large basin, put inside the temple for the priests to use for washing. I don't know if they actually got right into it and bathed, but it was certainly large enough for them to do so. It was pretty much a huge bowl of water.

So, how huge? Most of us don't know offhand what a "cubit" is, so that "Sea" might only be big enough to dip a Star Wars figure's head.

It turns out that a cubit is about 18 inches. We'll be coming back to that "about". So, this basin was about (there's that word again) 15 feet across, 7 and 1/2 feet high, and 30 feet around. That should explain why it got named the Sea (a term that appears several more times, capitalized as a proper noun, in the Bible).

Overall, though, you don't need to worry about what a "cubit" is. The conversion of cubits to metric or imperial measurements isn't the issue - the ratio between the distance across the Sea and the distance around it are what matter.

So, then, this passage seems to say that the Sea was 30 cubits around and 10 cubits across. In geometric terms, it had a circumference of 30 cubits, and a diameter of 10 cubits. But wait. The formula for the circumference of a circle is pi times the diameter. Pi is an irrational number (that is, it has an infinite, nonrepeating series of digits after the decimal place) a little larger than 3. It's been calculated out to thousands of places beyond the decimal, but most people are content to use the very reasonable approximation of 3.14159 in its place. Some use 3.14, but they're just lazy.

If we use the stated diameter of 10 cubits and multiply by pi the way the nuns back in parochial school taught us, we find that the circumference should be 31.4159 cubits, not 30. Conversely, if we start with the circumference of 30 cubits and divide to find the diameter, we find that it should be about 9.5493 cubits, not 10. So what's going on here?

To be fair, most people attacking the Bible over this aren't seriously suggesting that Bible believers think pi is precisely equal to 3. There's an urban myth that state legislatures have in recent years tried to pass legislation forcing pi to equal 3 on Biblical grounds, but it isn't true (which I didn't realize until researching this article).

Instead, those people are usually using this passage to "prove" that the Bible is unreliable on factual matters, usually to bolster their skepticism on other Biblical issues. See, for example, this article. The endgame argument tends to run along the lines of, "How come you understand that pi doesn't equal three even though the Bible says it does, but you still believe in the Virgin Birth just because the Bible says so?" For "Virgin Birth", feel free to substitute "global flood", "resurrection", "seven-day creation", any miracle, or anything else the Bible records that runs contrary to our everyday experience and modern, scientifically enlightened popular belief.

No matter what you substitute, the underlying claim is the same: the Bible cannot be trusted on factual, scientifically measurable matters. When the Bible and modern science conflict, the Bible is to be rejected. Refuting that argument in its entirety is well beyond the scope of this article (and probably beyond the scope of my knowledge and writing skills). However, we'll carry on with our look at pi, which may also chip away a bit at the argument outlined above.

The argument is that the Bible herein proves itself inaccurate and unreliable by stating that pi is equal to three. The implication of the argument is that the Bible is therefore untrustworthy in its entirety. I will not be addressing the implication (beyond to say that it requires a leap in logic which would bear further examination before acceptance), but I see three immediate responses to the argument about pi. Once again, in researching this article I found that others have made these same points before me. No matter.

These three offered responses are not mutually exclusive. The truth may lie in some combination of them, or indeed in some possibility I have not considered. I would only have people think before rejecting the Bible, and hopefully this helps.

First, the numbers indicated in the passage - ten cubits across, thirty cubits around - may well be approximations. This passage is a description meant to convey a visual image to the reader, not an engineer's blueprint. There is a large bookshelf a few feet to my left.If I say that it is five feet wide and six feet high, I would consider that sufficient to give you an impression of its size. I would not feel I had been proven a liar if you were to take a tape measure to it and find that it is in fact exactly 4 feet, 10 and a quarter inches wide and 6 feet, 2 inches high. This is the argument you will find in the notes of most study Bibles and commentaries - the numbers given here are rounded approximations.

A variation of this is closely enough related that I will treat it under the same heading. A good explanation of this variation can be found on this page at Purplemath, which also covers what I'll be coming up to shortly as number two.

There was no standardized definition of a "cubit". The word itself means an approximation, based on the length of a man's forearm from elbow to fingertip. As you can imagine, this isn't the same for all men. The cubits of the circumference may not have been precisely identical to the cubits of the diameter. Due to the rounding / approximation issue, this doesn't really matter, but I thought it worth noting.

Second, the measurement of 10 cubits "from rim to rim" could well be a measurement of the external diameter - that is, from the outside edge of one side of the basin to the outside edge of the other. As noted earlier, that would describe a circle with a circumference of 31.4159 cubits. (As earlier, don't get hung up on the units - the mathematical ratios are what matter.)

However, this huge bowl could not have had zero thickness, and certainly not negative thickness. Perhaps the 30 cubit measurement of the circumference was taken around the inside. Purplemath again does a terrific job with this argument, complete with diagrams, explaining that a second mold would be required for the inside of this basin, and that the 30 cubit measurement could have described that smaller mold.

A circumference of 30 cubits indicates a diameter of 9.4593 cubits. That leaves a discrepancy of .5407 cubits, or about 9.7 inches, to account for the thickness of the rim. Divide that in two for the two sides when measuring across, and you wind up with the brass walls of the Sea being 4.95 inches thick.

And, hey, what's this we read about the Sea in I Kings 7:26?

It was a handbreadth in thickness...

A "handbreadth" is another of those Biblical-era ad hoc units of measurement, and means pretty much exactly what it says: the width of a hand. The almost-five inches we calculated is a bit on the high side from what most study Bibles and commentaries say (although my own hand is considerably wider than the three inches they suggest, and I'm not a real big guy), but not unbelievably out of line. Besides, the rounding / approximation issue I described in my first response easily allows for that rim to come down (or go up...) by an inch or two. Assuming that's logistically feasible, of course. Not being an engineer, I don't know how thin those brass walls could get before giving out under the weight of all that water.

Once again, those who attack the Bible haven't read a few verses down from the verse they're basing their argument on....

On to my third possible response. To assume that a stated circumference of 30 units and a distance across of 10 units proves a mathematical contradiction is to assume that the object being measured is a perfect circle. I'll ignore the fact that physical reality contains no perfect circles, and no straight lines, for the sake of this argument. I like philosophy as much as the next guy, but I'm not going to use that kind of esoteric argument to try to get out of this. By "perfect circle", we mean one that is close enough to perfectly circular to treat it as such for all intents and purposes.

However, we have no reason to believe that the Sea was a perfect circle. "Circular", yes, meaning round, but once again we have to remember that this description was written to evoke a visual image of size (and therefore grandeur) and was not an engineering blueprint. It could just as well have been described as "immense, thirty cubits around".

I did some online research about the measurements. This reminds me why I love the Internet. Not so many years ago I would have pondered this for a few seconds, thought perhaps I should go digging through some geometry books sometime, shrugged, and forgotten all about it. Thanks to Google I was able to find some ellipse calculators and try out some numbers.

To get what I'm saying about the possible shape of the Sea, pick up a fairly rigid round plastic cup, and look at at from directly above. The mouth of the cup should look like a circle. Now, with your hands as close to the base as feasible, squeeze the cup until the mouth begins to distend slightly. The mouth will now be elliptical, but I wouldn't be shocked if someone was still willing to describe it as "circular".

That's what I'm saying about the Sea. It could have been slightly elliptical, and the 10 cubit measurement could have been taken across a wider section (in geometric terms, closer to parallel with the major focus than the minor). For that matter, the measurement across may not have been taken directly across the centre point, again affecting the results.

I found an ellipse calculator on 1728.com, and plugged some numbers into Calculator Two. If you enter a major axis of 10 and a minor axis of 9.08, you'll get a perimeter of 30.035 (remember, units don't matter, only ratios). Then, you can put those same numbers into the ellipse generator at Geek.Casaforge.com to see how close to a circle the results look. You'll find that "circular" is a fair description for purposes of the layman.

Feel free to play with those numbers - by adjusting the axes, you can get very close to a perimeter of 30 while the resulting figure looks very much like a circle. Remember too that the sheer size of the Sea would make a slight deviation from circular even less noticeable.

I wound up playing with those ellipse pages far longer than I should have. At least it wasn't as bad as that cursed online Spirograph simulator that held me in its grip many moons past. I almost wound up getting an intervention over that thing. But I digress.


While researching this article, I was pleased to see some sites where these (or similar) arguments were accepted to debunk the "the Bible says pi is three" trope. The "Atheism FAQ" at this site is a good example. Often people on one side or another of a debate, and I'm including my allies in this, are only too willing to throw any accusations they can find against the wall and see what sticks. Consider the 2008 presidential election campaign, when Democratic supporters were openly encouraging the spread of rumours that they knew to be false about Sarah Palin; as long as it made voters question her, the ends justified the means.

I don't like to see bad arguments getting used by either side of a debate, and the best way to prevent that is to police your own side. It's good to see anyone saying to their allies, "That argument has been refuted. Let's stop trying to use it and move to stronger ones."


Enough rambling. Here's a picture of my yellow Orks. I usually ran these guys as Mad Boyz.

The True Identity Of Jack The Ripper

Here's what my brain did to me today.


The identity of the Whitechapel murderer, popularly known as Jack the Ripper, has been a mystery for over a century. Many have tried over the years to prove that one suspect or another was secretly Mad Jack, the sociopathic killer. No one has ever conclusively proven their case.


Until now.


The true identity of Jack the Ripper was, in fact: Muppeteer Jim Henson.


It seems obvious now that it's been pointed out, doesn't it? I bet you feel really stupid and inadequate for not having realized it sooner. Suddenly, life makes sense.

What better cover for a methodical serial killer than to become a beloved family entertainer? No one ever suspects the guy with his arm up Kermit the Frog. Being born decades after the Whitechapel murders was a masterstroke of misdirection. Very clever, Mr. Henson! But not quite clever enough.

Consider: not a single prostitute murder in the Whitechapel district has been attributed to Jack the Ripper since Jim Henson passed away in 1990. There are also no published memoirs of Victorian-era prostitutes recounting tales of meeting Jim Henson. This can only mean that he left no witnesses. Not a single Victorian-era prostitute saw Jim Henson's bearded face and lived to tell the tale.

When this theory occurred to me, I first considered Walt Disney as the possible culprit. I was forced to abandon that path of thought because it hit a little too close to home. I'm really not sure about old Walt.

I have developed an elegant proof of this theorem, but unfortunately this Internet is too small to contain it.

Oh, all right, I'll try to cover the high points. You should be able to fill in the details for yourself. Suffice it for the moment to say that an irrefutable mathematical proof of my theorem exists, wherein we begin by letting X = "Jim Henson" and Y = "Jack The Ripper". Before long, X equals Y. It all hinges on the remainder left when 5 disemboweled Victorian prostitutes are divided by "Gavin McLeod".

As opposed, I suppose, to being divided by the Ripper's sharp instruments.

I would have liked to include some diagrams to support my theory, diagrams being an excellent way to sanitize complex mathematical concepts for the consumption of the unwashed masses. Maybe some pie charts. Everybody likes pie charts. But thinking about that got me to thinking about pie, so I went and looked in the kitchen only to find no pie, but I found some strawberry shortcake shells and some strawberries, but no whipped cream, but we had some whipped cream mix so I mixed that up and had a nice strawberry shortcake and by the time I was finished I had forgotten all about pie and Jim Henson being Jack the Ripper.


Enough rambling. Here's a picture of my son (at left, in camo in case his keenly honed survival instincts compel him to take cover in a wooded area of the mall), Pastor Derek and Homeschooler's daughter (in pink) and two innocent bystanders, mere moments before the situation turned ugly.

Quick Notes

Another assortment of quickies in lieu of a longer, better thought-out entry. I like doing these because they make me feel like Andy Rooney.

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I've been quiet lately for two main reasons. First, there's a big project going on at work, and staff were offered two hours of daily voluntary overtime. There weren't many of us who accepted, and even fewer who did more than a handful of days. I've stayed for the extra time almost every day since late January.

You wouldn't think that a mere two hours added to the end of the work day would have such an impact, but it definitely has. I haven't even mustered up the energy to do my tax return yet. Since I'm expecting a sizable refund, that's saying something.

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The second main reason for my radio silence: Gamehouse Sudoku. I normally play level 8, and my average time is about four minutes. I thought that the Automarks would make me lazy and actually atrophy any Sudoku skill I once had, but they only remove some of the grunt work. You still have to do your own thinking, which is the fun part anyway.

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Rebelangel has been posting lots of interesting stuff lately, trying to make up for the days she missed during her recent computer problems, and get back on track for her Blog 365 target. When there's nothing going on here, check over there. (When there is, check over there afterward anyway.) I'd be posting comments on some of her articles if I didn't have so many important video games to play. Maybe once the overtime wraps up (supposedly soon).

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Pastor Derek and Homeschooler's newborn twin boys are still in the hospital. This article has an update and some new pictures. Poor Henry looks so sad in the first and last pictures. He looks like he's far more aware of his discomfort than he should be at this age. Those are spookily wise eyes.

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On to less personal stuff.

The attorney representing child-killer Christopher Pauchay (remember him?) says that he was "surprised by the judge's assessment that Pauchay lacks insight into his behaviour."

Over fifty criminal convictions and decision-making skills that led him to carry his daughters outside in a blizzard to freeze in a snowbank, and his attorney is "surprised" by a suggestion that Pauchay might not think too much about his actions.

Maybe an IQ test should have been administered before the bar exam.

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The pictures I post here should start looking much better. I use Irfanview for most of my digital image manipulation, and recommend it highly. As I demonstrate on a regular basis, I'm not much of a photographer. My shots tend to come out with a yellowish tint. I've played with the colour balancing tools in Irfanview (Image - Enhance Colours), but all I've ever managed was to replace the yellow tint with a red or blue tint.

The other day, just for the heck of it, I clicked the Auto Adjust Colours option in that menu for the first time.

Wow! Never again will I mess around with manual colour balancing. With a single click, that button makes my shots look clear, colourful and vibrant. I'm tempted to repost some of my old shots with that option applied, especially the pictures of my wife's crafts. She does much better work than you would think from looking at my washed-out pictures.

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My local supermarket has Pringles on sale for $1.99 a can. The regular price, plainly visible on the sign announcing the sale, is "2 for $4.00."

I thought this was entertainingly idiotic and/or idiotically entertaining until I thought it over. You may laugh at a savings of a single penny, but think of the big picture: if you bought a million cans, you would save $10,000. When's the last time you had an opportunity to save that kind of money by buying vaguely potato-based snack foods?

I may need to rent a storage space.

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There are currently nine books in my "finished reading this - now write a Reading Log entry" pile. I finished some of them as long ago as last summer. I may not remember much about them now. I've also bought five more books in the last couple of weeks, all of which will join that pile soon enough.

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My MP3 player, which I listen to much of the day at work, has had Bob Dylan's Infidels album on it for a couple of weeks now. Since I normally rotate its contents every day (usually two albums / concerts daily), that means it's holding my interest pretty effectively. I had never heard it before. A friend at work is a big Dylan fan, and after we discussed the three "born again" albums, he brought it in for me. I may write an entry on it in the future. I already have draft notes.

Today's other selection was "Beatle Mash" by "The Liverpool Kids", a no-budget and no-name exploitation released circa 1964 to cash in on Beatlemania. It's not awful, but it's certainly not memorable. Generic early sixties R&B, with one Beatles cover (She Loves You) and one "original" that's such a knockoff of "I Want To Hold Your Hand" that the producers would surely have been sued for plagiarism if the album had been noticed at the time. It's a closer copy than "My Sweet Lord" is of "He's So Fine" - and that's pretty darn close. This album won't be held over.

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In the mid-eighties, the writers of Spider-Man comics did a storyline that tied into the much-maligned Secret Wars II (which rocked my barely-pubescent world, despite being sheer crap in retrospect). The Beyonder turned an entire Manhattan office building, and all its contents, to solid gold.

Hijinks ensued as Spidey had to rescue its occupants and neighbours (a solid gold building apparently cannot support its own weight; I'm sure the writers had engineers check all the math), the Kingpin tried to steal it, and finally the government stepped in to confiscate it and get rid of it (I don't remember how, and I'm not subjecting myself to re-reading a Secret Wars II tie-in).

The writers showed some real economic insight here. It was explained that the sudden ex nihilo introduction of all that gold would destabilize the world metals markets, and ultimately the entire global economy. You can't just suddenly flood the market with a previously scarce commodity without doing far more harm than good. An inflationary spiral, followed quickly by economic chaos and collapse, invariably ensues.

This goes for a government introducing large amounts of previously nonexistent currency into the economy, whether by actually printing bills (remember Germans needing wheelbarrows of cash to buy bread under Hitler's economic stimulus package?), or by giving large amounts of money that only exists as numbers on a screen (most "money" doesn't physically exist anymore, it's all abstract) to... well, anyone.

How sad is it that Spider-Man writers of the mid-eighties were smarter economists than anyone in the Obama administration?


Enough rambling. Here's a picture of my son doing his absolute best to hold still. Check out that sharply-defined colour!

Education Confrontation Part 3

It's been roughly three-quarters of forever since I posted part 2 of this series, but I might as well circle around to wrap it up. For those who came in late or don't obsessively memorize every word I spew onto this page, here are links to the earlier parts to refresh all of our memories: Part 1, an Interlude between parts 1 and 2, and Part 2. The Interlude was a response to my first (and so far only, to my disappointment) silly flame comment. As far as I can tell, the person who wrote the comment did so mainly to make me chortle. I especially enjoy the bit where they claim to be a small-government libertarian while simultaneously calling for my arrest on charges of disagreeing with their opinion.

A lot of troubled water has passed underneath this particular bridge since part 2. In fact, things got really weird right after I finished that part, which is a big chunk of why the series got derailed. Perhaps I'll go into more detail later. A lot of it involves things that would be news to some friends who read this, and I'm not sure whether they should find out by reading it on here. For now, I'm going to try to carry on as though much of what has happened over the last couple of months didn't. (The new developments are inconvenient, not dire, but they have necessitated some rearranging of short and long-term plans.)

I had always expected to wind up gritting my teeth and going along with a lot of nonsense from school system bureaucrats, for most of whom "useless" would be a step up. Now I pretty much know that's exactly what's going to happen. For various reasons, our son is definitely entering the public school system this fall. Extensive preparations are being made at their end for his arrival.

My philosophies have not changed. However, as almost always, I will be making some concessions and compromises. It just means a little more ulcer medication than usual. (One of my favourite moments from NYPD Blue was an episode where Andy Sipowicz glared at a suspect for a while, then rasped something like, "I'm gonna have a headache tonight because I didn't beat you." I can relate, almost anytime I have to deal with almost anyone.)

Sidebar (first in a series): I have long understood the word "compromise" as being a euphemism for "a partial solution that won't make anybody happy." I generally don't see it as a good thing.

To recap my earlier parts (and make a few underlying principles more explicit), the school system (meaning primarily the teachers and bureaucrats) has no rights beyond those specifically delegated to it by parents. I'm not qualifying that in any way. No "should", no "in my opinion". Homework is no more acceptable than your employer ordering you to do some work at home, on your own time, for no pay. (I understand that's effectively the situation for, ironically, schoolteachers. Maybe the fact that they put up with it explains why they think it's acceptable to do to others. They shouldn't, and it isn't.)

Detention, without explicit prior parental consent, is unlawful confinement. As is any "requirement" that students stay on school property beyond classroom hours (i.e., not letting students who live near the school go home for lunch) if their parents have given them permission to leave. Either one should legally be treated the same as any other hostage-taking.

An awful lot of what's wrong with modern society comes from petty bureaucrats overstepping their authority, and being allowed to get away with it. I'm as guilty of that second part as anyone. I exemplify what Paul said in II Cor 10:10; I'm much easier to get along with, and much likelier to defer to authority (either legitimate or delusional) in real life than you'd expect from reading this. I'm pretty pragmatic in everyday interactions, and it's often easiest to let someone think that you actually care what they think.

This, of course, makes me part of the problem.

Each time one of these dime-store tyrants throws their imaginary weight around and doesn't get knocked on their metaphorical backside for doing so, it emboldens them. In psychological terms, it reinforces the behaviour. Writing this makes me realize that maybe I should stop being so agreeable with them, for the greater good. The sooner these petty bureaucrats (words that I need to use together all too often) understand that they are small cogs in the big machine of society, and that their personal opinions are worth the exact midpoint of "jack" and "squat" in the course of performing their duties, the better for all of us.


Sidebar of an example (and a True Story From One Of My Jobs): once upon a time, I worked in a government office where photos of citizens were handled. I can't get too specific for various reasons, not least of which is the protection of my secret identity. One day, the staff encountered a client photo that some of them felt was inappropriate. Let's say, the client was wearing a shirt with something bluntly critical of the government written on it (not precisely accurate, but close enough).

For the purposes for which we needed the photo, it didn't matter at all. It would be cropped in such a way that the fellow's protest would never even be seen by anyone except a few of our staff. However, some people on our staff wanted to reject the photo, and there were suggestions of further retribution. Some wanted to delay his paperwork unnecessarily, and one or two even speculated that perhaps the client could be charged with something.

I was disgusted by the attitudes of my co-workers. You could argue that the client was being rude, at worst, and there's no law against that. We had absolutely no right and no basis to so much as question his photo. I argued on his behalf, but ultimately a supervisor walked by and got involved in the discussion. After glancing at the photo, the supervisor decreed that it was unacceptable. The client was to be told that he needed to have another one taken. I asked for a legal basis for this, but the supervisor refused to discuss the matter (justifiably, in a way - I answered to the supervisor, not the other way around).

This is a textbook example of what I'm talking about. The personal opinions of some insignificant cogs in a huge bureaucratic machine were allowed to override all legislation and regulations. That should never, ever happen. The client's photo was perfectly acceptable by the letter of our regulations, and that should have been all that mattered. I'd like to say that was the day I realized I wanted out of there, but I had already realized that and become a self-loathing civil servant well before then.


In my experience, the school system in even more infested with this mentality than most organizations. Perhaps it comes from working with children, who are even easier to bully than most adults. However, school system employees tend to forget the difference when the time comes to deal with parents. They quickly forget their role: they are there to support the parents, not the other way around. My wife and I are in charge of our child's education. The school system is there to assist us toward that end. Their suggestions are welcome, but we are not here to do as they say.

So, back to the main narrative of this series. We return now to April 8, 2008, in a classroom where the school principal is telling a group of parents what to expect when their children start school in the fall. Part 2 left off with my evisceration of their monthly "award assemblies".

The next thing in my notes (part of the problem with the long delay is that my memory has largely faded, so the notes become more important) is that the principal used the phrase "statistics show that" far, far more than I would have liked. I'd estimate that she said it 14,256 times - a number as valid as most reported statistics.

Along the way to my degree in psychology, I took a few courses in "research design and analysis", colloquially known to the students as Stats. There was more to the subject, especially in the advanced courses, than statistical analysis - although there was a lot of that as well.

I don't remember offhand anymore how to do a T-Square Analysis, or how to run a multifactor ANOVA. However, remembering exact procedures is not the point of education (for most fields, emergency room physicians being a notable exception). Instead, students should generally learn principles to apply, logic, and methods of researching specific questions. Memorization of specific methods, while useful in the short term, is not a good long-term strategy because methods change. Consider for a moment someone who got a degree in computer science in the early 1990s, but hasn't glanced at a computer beyond playing an occasional game since. How valuable would their dated knowledge be these days?

The main principle I took from my Stats classes was that statistics can be manipulated to say whatever the speaker wants them to say with trivial ease. It's actually extremely difficult to craft a survey, on any topic, to return reasonably objective results. That being the case, while statistics interest me, I don't put a great deal of stock in them. I've often said that I could design a survey on any issue that would return results showing that an overwhelming majority of the population feel however I want them to feel. The sponsor of research can essentially always get the results they're hoping for. The hard part is avoiding that problem, in the case where you want research to be broadly useful, as opposed to just supporting an agenda.

So, getting the impression that the principal seems to be a disciple of whatever USA Today says on their front cover pie chart doesn't excite me. Maybe she's very discerning, or very lucky, and winds up looking at useful research (which is rare) instead of agenda-bolstering nonsense (which is rampant). Maybe.

Second, when it comes to dealing with individuals, statistics are never more than starting points. Remember what I said about bullies back in part 2? Most of them do not have low self-esteem. However, some do, and trying to treat one who really does have low self-esteem with methods meant for the majority is only going to make matters worse. That principle applies to everyone that is dealt with on an individual basis for any reason. Children are not widgets, and running a school like an assembly line is doomed to failure, as are most students of such a school. Everyone deviates from the norm somewhere along the line.


Now, frustratingly (probably more so to me than the reader), I'm ending this part here, having dealt directly with a whopping one point that came up during the meeting. It's late, and I want to sleep. I'm already breaking my general rule of only posting on spiritual issues on Sunday, since it became Sunday in my time zone about an hour ago (although there is some Bible in this installment, if I wanted to be really pedantic). Now that I'm back onto this series, hopefully it won't be over two months before the next installment.

Enough rambling. Here's a picture taken out the car window.

Get Jeff Foxworthy On The Phone

I'll probably feel bad about this one later, but for now I'll go with it being funny.

It's been said that comedy equals tragedy plus time. I've heard that idea from different sources, but Internet consensus seems to give Carol Burnett the original credit.

I'll agree with that formula. I prefer to define the value of time as "1". Decide for yourself on the unit of measurement.

Anyway, here we go. My favourite line, the one that sums it all up, is bolded:

WINNIPEG - Police are citing alcohol consumption as a factor in a tragedy in which a woman allegedly killed her sister-in-law and injured two other family members by driving over them while they slept on the ground outside her home.

RCMP said several people were at the home in Berens River at the time of the incident Saturday afternoon, and they had been drinking.

"This is an event that involved a lot of alcohol consumption," Sgt. Line Karpish said Sunday. "There was a gathering at the house at the time of the incident."

While driving out of her yard in a Pontiac Tempest, the woman ran over the three people, RCMP said.

So many jokes, so little time or conscience. You'll be able to make up plenty of your own, I'm sure. Let's try swinging at a few softballs:

-If you've ever gotten really drunk and run over some relatives passed out on your front lawn, you may be a redneck.

- If you've ever gotten really drunk and passed out on the front lawn, then a drunk relative drove across the lawn and ran over you, you may be a redneck.

- In defense of the folks who were "sleeping" in the yard, that's where this family keeps most of their furniture.

- The driver had to use the Tempest, because she couldn't get either of the Gremlins down off their blocks.


I love that the story specifically names the model of car, as though it's relevant. The marketing folks at Pontiac must be thrilled. Sure, getting product placement in an A-list movie is one thing, but this is the sort of publicity that money just can't buy.


Enough rambling. Here's a picture of something else that I find funny but that somebody, somewhere, will probably find a way to find offensive. Maybe even blasphemous. (I can hope.)

A True Story From One Of My Jobs

Hey, Kids! It's time for a True Story From One Of My Jobs!

We'll be going back a few years for this one, to a job I used to have but don't anymore. I worked in a small department that provided technical support to the rest of the operation. We got a call from a middle manager who had been asked by The Boss to provide some numbers. No problem. An important part of this middle manager's job was recording and analysing statistics for their department and providing all sorts of reports upon request from up the chain.

So, they did what they almost always did in these situations: called the support crew to come give them the numbers they needed. One of my colleagues - we'll call him Alex - agreed to go up and help them, mainly because he's much nicer and more patient than I am.

They needed to give The Boss the number of e-mails that come into a particular mailbox each month on average. Here's what took place:

Middle Manager: "The problem is, we don't have numbers broken down per month. We only have the annual total for each year."



(I'll give you all a minute to digest that.)



(There. Done?)



Alex, demonstrating remarkable restraint: "Well, just divide that by twelve, and you'll have the monthly average."

The grateful manager hauled out a pencil and paper and began dividing the year's total by 12, long-division style. Alex interceded and suggested that a calculator might be easier (and, although he's too tactful to say so, infinitely more likely to arrive at the correct answer). Then -

Middle Manager: "OK, that's the average for last year. Now how would we do that for the year before last?"

Alex, being kinder than me, didn't ask whether that year had more or fewer months to divide by.


We also realized, after he returned and told me this story, that he never explained why the annual totals had to be divided by twelve. I feel a sinking certainty that since that day, any time this manager has needed to calculate any average, they've divided whatever number they have to work with by twelve......

Enough rambling. Here's a picture of my son's feet after a day of playing outside.