full moon

The moon was almost full last night, and sitting right up in the sky outside my living room window in a clear sky, so I had a good view. When this happens I get amazed that I can see variation in the moon’s surface.  I was trying to think this thought: “Amazing.  It’s ______ miles away and I can see all the _______.” But I didn’t know how far away it was or what the variations I could see actually are.  This is the sort of thing I think I am supposed to know but don’t.  Then I remembered that if you don’t know something you would like to know, you can find it out.  This is an obvious thing, an advantage of the brain plus books plus other people plus the internet, but I am constantly not bothering to do anything about it.

So I looked this one up.  (On the computer.) First up was Google’s answer – the moon is 238,900 miles from earth. I started thinking about how to really get how far that is, but then I spotted another search result three items down in the form of a question: How far away is the moon? I liked the sound of that, linguistically speaking, so I followed the link to the NASA SpacePlace page, which appears to be set up for younger people than myself.  Such pages can be excellent for learning information that one thinks one is already expected to know, because it’s (sometimes) offered in clear and plain terms, though one might also have to read around some condescension, which is too bad.

NASA’s explanation, not surprisingly, is a little more detailed and nuanced (accurate) than Google’s.  The distance between the earth and the moon changes, so it can be expressed as an average or as a measurement at a particular moment in its orbit (which is not a perfect circle, hence the variation in distance).  According to NASA SpacePlace, the moon is 238,855 miles on average.  At its closest, 225,623, and at its farthest, 252,088.

I had to stop myself from reading much more just then because when it comes to the moon, several very cool words come into play (apogee, perigee, penumbra, et. al.) and as a result I knew myself to be at risk for spending the rest of the day becoming a moon expert (relative to how I’d been night before) and I had other things people were counting on me to do.

All of that is to say I was glad to be reminded that a few seconds spent asking a question and looking for an answer is enough to put things in perspective, and when things are in perspective, universe-wise, it’s hard not to be amazed and inspired and re-energized for whatever earthly toil one has assigned one’s self, even in a terribly tumultuous time.

a few things about smallness

The other day I thought maybe it might be time for me to try calculus again.  I didn’t take it in high school the way I might’ve if I’d been born a bit more recently, because there wasn’t a year left for it after 9th grade algebra, 10th grade geometry, etc.  And then I didn’t take it in college because, I think, I was tired.  Or didn’t get along well with the teacher?  I can’t really remember.

But there it is, out there, this un-learned mass with the fancy name. One of the young people I know, with whom I have been doing math for several years because she likes it and doesn’t “need” a tutor but likes having one, is nearing the end of pre-cal.  She and I are really more like math buddies than tutor and tutee.  I help her when she needs help, but a lot of the time we find interesting problems to try to solve together. Now, there she is, about to start calculus, and not only will I not be able to help anymore, she’ll likely be too busy with her calculus to do math with me much anymore.

So I’ve been brushing up on my pre-cal, looking for gaps and rusty spots, and considering  calculus texts.  I’m a Martin Gardner fan, and my library had a copy of Calculus Made Easy, the Silvanus Thompson text with revisions by Gardner, so I tracked it down in the 515s and opened to the table of contents. Thompson’s first chapter is called “To Deliver You from the Preliminary Terrors.” I immediately texted the words to my mom, who is still terrified by most math and went to all sorts of lengths to avoid passing her fear on to her female child.

I didn’t have terrors, per se, but I can’t resist this sort of language.  In this chapter Thompson delivers us from our preliminary terrors by explaining that to learn the two principle symbols d, meaning a little bit of, and ∫, meaning the sum of all the little bits. (He mentions the fancy language as well, but uses the plain to explain.) The second chapter is called On Relative Degrees of Smallness. I will admit that I was so enamored of this title that I had to stop reading for awhile. But then I resumed.  Here’s how the chapter begins:

We shall find that in our processes of calculation we have to deal with small quantities of various degrees of smallness. We shall also have to learn under what circumstances we may consider quantities to be so minute that we may omit them from consideration. Everything depends on relative minuteness.

Here I had to stop reading again and stare out the window.  Who knew this was going to be a philosophy book?

I recently had occasion to look up the word pico.  I have always been partial to words that mean lots of different things and pico is, among other things, the Spanish word for peak (or beak), an island in the Portuguese Azores, an acronym for a method of  locating relevant clinical literature, and a prefix denoting one trillionth. 10−12.  0.000000000001. Very, very little.

The other night I heard about Erika Christakis’ new book The Importance of Being Little.  This book is about very young children, and the way the world view of such people could inform the way we older bigger people interact with them.

This is an awful lot about smallness in one week, I’ve been thinking.  A little hard to ignore.  Also, my father, who is recovering from a massive stroke, cannot remember things I have just told him.  To find the patience and presence to dial back a lifetime of conversational habit in order to sustain an exchange with someone who can cast backward and forward only a few moments is at once an enormous thing and a very very small thing.

I’m not sure how far I’ll get with the calculus, but I can’t help feeling that it’s a terrific thing every time we consider smallness in great depth.


I went to a concert yesterday in a church sanctuary. The concert was unrelated to the business of the church, but the materials (bibles, hymnals, etc.) remain in place no matter what is happening in the space (services or otherwise).  A stack of cards in a box on the back of each pew read “Welcome.  This card may help you communicate a need or provide information.”

I had this thought: What if we gave cards like these to children in school? What if young people had an ongoing opportunity to communicate, on paper? When I read the card, I got the impression that my experience mattered, that I was being invited to participate.

I thought about the young people I know and wondered what each of them might write on such a card.  The first person who came to mind is in sixth grade. He might write “Most of the time I’m confused when we’re doing math.”  Another would likely say something like “I’m really interested in invasive species; can we learn more about that?”  Two high school students I know might say “Is there any way you could punch holes in our handouts before you give them to us, or keep a hole punch in the room so we could punch them ourselves? That would help me keep my binder from getting so messy.”  Another might say “I’m really not trying to be a brat, but I’m bored, and it’s so frustrating when we spend so much time going over the homework that is exactly what we just did in class the day before.”  Given the chance, I think they’d tell us all sorts of things that would be helpful to know.  Things that would help us understand their behavior, things that would help us provide the best possible support for them, things that would help us know them better.

My guess is that there are some teachers and schools who have something like this in place, some way for students to communicate with teachers and administrators that does not require trying to get an adult’s attention verbally, or reveal his or her need or information to all those within earshot.   But in general, I think we expect that young people will speak up if there’s something they need to say, and I think mostly they don’t.

Some might abuse an “opportunity to communicate a need or provide information” in an anonymous fashion, as one quick look at the internet will tell us.  It might be worth the sifting and sorting it would take to manage that, though, if it would make it possible for us to hear more from  young people who are sitting quietly not communicating with us, and, too, many of the ones who aren’t sitting quietly but aren’t going to say theirs out loud either.

Even if none of them wrote anything down, I think they’d get that message that I got when I read the card sitting there in the pew of a church I don’t even attend – that if I was there, I was invited to communicate and thus to participate.


On Saturday we had a visit from four friends – my college roommate, her husband, and their two sons, who are six and three.

The three year-old and I found, in a box of old games, a cribbage board.  He was enamored of the board when he saw the hidden compartment which houses the little pegs.  He also liked the surface, with a track corresponding to each of the three peg colors in the compartment.  He decided that we would each choose a color and then move the pegs around the board according to what we each rolled on a  six-sided die.

He is mostly confident at counting up to six, but has trouble distinguishing between the four and the five on a die.  I was reminded, when I noticed this, of how much math learning will take care of itself in the course of dice games played with young children.

I didn’t have to do any instructing about this four and five thing, though I’d have been willing to if I’d been asked. I just counted out four each time I rolled a four, and counted out five each time I rolled a five.  When he wasn’t sure which he’d rolled, he’d say one or the other, lifting his tone slightly at the end, just enough of a question for me to confirm or correct whichever he’d chose. After a few rounds of this he stopped getting the two confused, and stopped reading the numbers as questions, because he knew he had figured it out.

the sorter is broken

I asked a high-school aged friend if she’s thought about what she might want to study in college (she’d already told me she wants to go) or what she might want to do after college.  She hesitated, and then said that she really likes science but she’s terrible at memorizing things, so probably she’ll do something with history instead. She said she’s especially interested in environmental science and issues related to climate change.

You could think of this as a teaching or curriculum problem, but I don’t think it really is.  I think it’s a sorting problem.  Or a problem with how we’ve taught ourselves to imagine we should be sorted when it comes to ability and suited-ness to particular tasks and professions.  And also a problem with how we think about what our brains are best used for.

This young person thinks that her difficulty with memorizing will exclude her (or should exclude her) from a career in science.  She has classmates who memorize easily, and they are the ones with high scores on tests and in courses. So on paper, on transcripts, if it’s good grades that tell us about someone’s aptitude for a particular area of study, we can see that her quick-memorizing classmates are the ones destined for careers in science.  Yes, many a good teacher will tell you that if you don’t emphasize the memorization you can show something different with how you grade, but for most students in most schools, information recall is a big factor in grade determination and a teacher who makes it otherwise is swimming upstream and trying to pull her students along with her against a strong current.

What would we have to do to make it otherwise?  First we’d have to decide whether we believe that a scientist must be able to keep on hand a multitude of data.  Must one?  I’d guess not, at a time when it’s possible to put a handheld device with endless data in the hand of any professional anything.  Wouldn’t it make sense for the first qualifying characteristic for a career in science (or any participation in science) be an interest in participating, and then perhaps the second an interest in and capacity for problem solving and analytical thought?  Science once required extensive memorization, but it doesn’t any more, and we’ve got big enough problems, and many enough problems, that the more solvers we can get on them the better, it seems to me.  To exclude the ones who can’t memorize stuff as well as some others can memorize stuff seems unwise.  Not to mention that the good memorizers might be put to better use elsewhere especially if they’re not interested in the careers and occupations that their memorizing might qualify them for and point them in the direction of.