This latest math-y video from Matt Parker may also appeal to word people…
Here goes Matt Parker again, doing fun things with math.
Today is 4/9/16 (in places that put the month first) which means three perfect squares in a row: 22, 32, 42. Last time that happened was January 4th, seven years ago, and next will not be until 2025, in September on the 16th. After that we have to wait for a new century again because we don’t have enough months to make it work for 16/25/36.
For everyone who lives everywhere else, the first and second dates this century would be 1 April 09 and 4 September 16 instead, and the third one wouldn’t happen for the same reason there won’t be a 16/25/36 on our calendar. Not enough months to write 9.[16th month].25.
The moral of the story is that we get one extra perfect square series by naming our days with month first and day second. Not sure it’s worth the confusion created by numbering differently from how everyone else does, but that’s for another time.
Here it is without all my nattering:
1/4/09 (January 4th), 4/9/16 (April 9th), 9/16/25 (September 16th)
1.4.09 (April 1st), 4.9.16 (September 4th), –.–.– (9th day of 16th month in 25th year)
If you like really big perfect squares, you can smash the days, months, and years together. I’m going to work on having more capacity for conceiving of these giant ones. In the meantime, fortunately, someone else posted about that.
This is sort of another plug for Ed Zaccaro, but also an observation about language and dignity in the context of math.
I posted a while back about Ed Zaccaro’s Challenge Math series, which I think was written for kids who needed more than the classroom could provide, but also seems to work well with kids with all sorts of different relationships with math.
This week I was surprised when one of the eight year-olds I work with asked specifically if we could work in the Zaccaro book. I’d done a quick bit of one page with her once before, and she was OK with it, but tends to balk at anything she has to stop and think about, anything that requires reading as the Zaccaro problems do. This is not because she doesn’t like to read or isn’t a good reader. She is. But as a math student, she tends to balk at reading. We’d mostly been working on quick little problems I’d chosen for the purpose of letting her see that she is more capable when it comes to math than she has learned to believe.
She leaned right in as soon as I opened the book in front of her. She remarked on the illustration of the large sweating ants and then started reading the first problem on the page. She sped through three in a row, understanding exactly what the problems were asking for and doing much of the computation in her head. This is exactly what she doesn’t do when she’s interacting with her Pearson pages from school (which, as far as I can tell, were written and issued very quickly after the Common Core standards went into effect). She’s constantly choosing operations that don’t fit the situations in the word problems.
I’ve been frustrated with the language of many of these problems ever since the materials were issued, but in watching this child with the Zaccaro problems, I realized something new about how they are tripping kids up. The Pearson word problem language is often ambiguous, which is terribly troublesome in the context of math, but it’s also just plain dull. For kids who are readers, accustomed to sentences written with care and intention, poorly written problems are not just potentially confusing. They’re an insult to the sensibilities of the literate child. The literate person.
I realized that what this student has been doing with her classroom math work is to scan each word problem and attempt to match it with a procedure. If it looks like the kind of problem for which one should divide, she’ll divide. If it looks like you have to trace your finger across the table shown and fill in a number according to a pattern, she’ll do that. No matter that often the problem is asking for something slightly different. In order to notice that, she’d have to subject herself to the agony of reading the dull text slowly. As a result, her actual ability and number sense is obscured. In a situation in which the reading is in and of itself engaging, she is freed up to think fluidly and flexibly about what there is to do with the numbers.
What the Zaccaro book does for her (that the Pearson book does not) is offer something. That sounds too simple to matter, but I don’t think it is. An offering concerns itself with presentation and connection, free of attachment to result. It is the fundamental unit of authentic human exchange. It says “Here. This is for you to have and use if you choose.” It grants dignity, agency, autonomy. In the course of our deep but desperate commitment to educate, we’ve moved away from offering toward mandate and decree, often undermining our ability to pass on what there is of useful knowledge and skill to our young.
The good news is we can resume a culture of offering any time we want.
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.
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.
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.
I was on the phone with a mom today who told me this story about her son, who thinks of himself as a lousy student who can’t do math and isn’t much better at other things. She told me:
“I was filling out the form [for school] and I asked him what he thought his new teacher should know about him. If there were one thing he’d want her to know, what would it be? He sat there for a minute, looking out the window, and then as he went back to what he was doing he said ‘Tell them I’m kind.'”
This child knows that kindness is not the currency of schooling. He knows that it isn’t what anyone will be measuring when he gets there. And maybe he thinks that kindness is all he has to offer, in spite of its relatively low value in the eyes of the institution (as indicated by the fact that it doesn’t appear on any report card he’s ever seen or heard of).
Or maybe he thinks it’s important, and thinks that while his past performance on timed math tests and the like suggests he doesn’t have what school is looking for, he does have something that matters, or something that should.