Afterthoughts, Afterlinks, Resolutions, and Thanks

My most recent post, about math mentors and math fun, was the 25th on this blog. The calendar year is about to turn; I’m a little less than halfway through my year to think it over. Time for a mixed salad of quick thoughts, including some resolutions.

More math fun

First, it turns out that 2013 is not prime. All year I’ve wondered. Finally, this morning, I started scratching calculations on the back of an envelope. Then I went to the web to double-check, and found a prime factor calculator. That confirmed it, just in time:

goofygraphics2013I also checked 2014. That, too, is a product of three primes, but I’m not telling. (You know right off the bat what one of those primes has to be.)

After reading the previous post, Kate Keller said some really nice things, including, “Wait! You left out the birthday ritual!”

Kate’s remembering something we did whenever a birthday happened in my class. I started by writing the child’s new age on the big whiteboard: 11, or 12, or occasionally 13. Then I’d ask, “What can we say about the number —?”

Students responded in a variety of ways:

  • with cultural uses of the number. (“It’s a dozen!”) (“Some people think it’s unlucky, but they’re crazy…”)
  • with expressions that equaled the number: 3 x 4 = 12, or 14 plus -3 = 11, or much more complicated expressions coming out of our experience with Lloyd’s Game (described in the previous post.)
  • with a magic number sequence that started with the number and returned to the number.
  • with words that describe other properties of the number: it’s odd (or even); it’s prime (or composite); it’s a palindrome; it’s deficient or abundant or perfect.

These various statements, written on the whiteboard, both documented learning and provoked it. Although we focused on the same numbers again and again, the activity was repetitive only in the way ritual has to be repetitive: a pattern similar in every iteration, but never actually identical; a shared dance in which roles can change and change again; a bowl or basket or web for both familiarity and innovation.

If I forgot to include the number ritual in our celebration of someone’s birthday, or if we ran out of time before dismissal, the kids insisted that it be carried over to the next day. Remembering my students’ affection for the ritual, and remembering the way every student participated, I feel like I’m holding some important key to who they were, and are; something hard to put into words; a treasure.

For still more math fun, check out the YouTube channel of Vi Hart. Here’s a link to one of my favorites, the first in a sequence of three about plants and the Fibonacci sequence. “Ow!” one of my younger students said. “My head hurts! Play it again!”

vihartfib

More My Place

I’ve been tickled to have the posts about My Place get a steady trickle of hits from Australia, so I did some behind-the-scenes backtracking. In the process, I found a wonderful collection of material relating to Nadia Wheatley, with an author interview, curriculum plans, and reviews of some of her other books–and a link to my own post about My Place, down towards the bottom. Great stuff!

If I were teaching right now…

I would read aloud The Higher Power of Lucky, the first in a series of three novels about a girl named Lucky in a town named Hard Pan, in the Mojave Desert.

higherpowerofluckyb

Living in a very small town, Lucky has memorable friendships with both kids and grown-ups. She eavesdrops on twelve-step anonymous meetings, hoping to hear the advice she needs. She hopes seriously for an afterlife, because there are some questions she would like to ask Charles Darwin. (She has a dog named HMS Beagle.) She’s cranky and impulsive and imperfect and worth a million dollars, and she’s part of a new sub-sub-genre of realistic contemporary fiction for young adults, in which characters think about biological evolution and what it means, and interact sympathetically with adults who can’t or won’t.

“If” thought # 2: I would figure out how a class could use the latest book by Alice Roberts, the charismatic anthropologist and medical doctor who narrates a BBC video series (which we did use in class) called The Incredible Human Journey.

alicerobertsevolutionbAlice’s new book (we pretend to be on a first-name basis with her, in my household), published by Dorling Kindersley, is called Evolution: The Human Story.  It uses narrative, model reconstructions, photographs, illustrations and charts, to take the print medium’s slower-paced (but thrilling) look at the history of our species, starting with the Big Bang. Such a rich resource for a class to use!

A third “if” thought: I would explore the idea of privacy, which matters a lot to 11- and 12-year-old people, and keeps coming up in the news.

Resolutions

One of my most faithful readers wants to know why I haven’t yet written about some teaching and learning that was central to my teaching life:

  • about the evolution of life in general, and about human evolution more particularly;
  • about animal behavior and archaeology and the history of technology;
  • about immigration, both chosen and involuntary, in the history of our country and our communities and families;
  • about Islam and the Arab world and the history of Arab Spain;
  • about The Voyage of the Mimi, both the first and the second;
  • and about the making of Voyage to the Sea.

Instead of writing about evolution, I guess, I’ve been evolving. (I know; I’m using the word in two of its different senses.) Somehow I’ve had to work up to those topics, and also work down with them. They’re all so huge for me, giant human artifacts around which I’ve spent all these years crawling, like an ant in the jungle, climbing up and looking around whenever I felt brave, or whenever a student was nudging me onward.

However, I’ve just made that list. I’ve included some of it sideways, in this mixed salad post. I’m pledging myself to figure out ways of exploring those giant thoughts in 1000 word packets, before my year to think it over is over.

I welcome, and probably need, suggestions from readers who shared those themes with me as student or parent or colleague or cheerleader. If you were writing this blog, how would you tackle all that big stuff? Just askin’.

In an activity so solitary (except for the joyful throng of co-conspirators in my memory), tiny encouragements from the rest of the world mean so much! A quick note in an email, a side comment in the aisles at Colella’s, a post on a website generated on the other side of the planet, devoted to a much-admired author–each of these remind me that I’m really doing this, and parts of it matter to other people. Some of you have recommended the blog, or a particular post, to friends and relatives and colleagues, or on Facebook; some of you have written comments on the blog itself, invariably thought-provoking, nudging me and lifting me forward. For all that…

goofygraphicsthanksRecently, my daughter has been sharing a website or movement called Lean In, which encourages women to lean into their ambitions, to overcome fears and take risks, with each others’ support. I take a big breath and “lean in” every time I publish one of these posts, and I’m inviting you to lean in with me, women and men (and girls and boys)–whatever that may mean for you.

Math Mentors and Math Games

Early in my graduate education, I took a Lesley University summer school course about teaching mathematics, with a genuine, fresh-from-the-trenches middle school math teacher, Lloyd Beckett. Authentically–and contagiously–he had come to believe in the power of math conversation, and in the rich gifts students with different approaches could offer each other.

Lloyd’s course woke me up as a math teacher and as a mathematician. For years I had assumed that my relatively decent math grades rested completely on my ability to memorize. As it stood, that was largely true. When I did particularly well on one of the New York State Regents exams, my teacher, Augustus Askin, whom I adored, looked at me and said, “How did you do that?” Although I don’t think he suspected me of cheating, he had seen the puzzled look I often wore in math class.

Memorizing was okay for the test, but the effects never lasted very long. Real understanding, for me, required experiences that math class rarely offered–that I couldn’t even imagine math class offering.

On the other hand, in secret, generating that puzzled look, I’d spent years figuring out my own approaches. I could hold onto math concepts, and work with them comfortably, if I experienced them pictorially or concretely, or told stories about them. This was in a time, though, before math manipulatives, at least in my country schools, and before the wonderful math videos I was able to use with my own students, decades later.

There were exceptions. A little girl for whom I babysat had one of those balance toys with numbers weighted to add or subtract properly. If you hung a 5 and a 2 on one side, and a 7 on the other, the balance came to rest with the pointer in the right place to mean yes.

Here’s a sample of a similar balance still on sale.

math Plastic-Bear-Shaped-Digital-Balance

Other, purer versions make more sense for older kids, but I spent a lot of time playing with that balance, savoring it. It was what I needed.

I’m also stubborn, and I hated subtracting. All on my own, with no support from the rote-memory approach in school, I had figured out a way of subtracting by adding, doing a sort of mental algebra: what plus 5 will equal 7? Or what plus 9 will equal 17?

In my earnest little heart, though, I suspected that I was cheating. I thought I was making up for not being good at math.

Years later, when I spoke with parents at math curriculum nights, I sometimes called myself a “born-again mathematician.” Teaching math with new math tools and toys and approaches, and with new respect for many kinds of math minds, I found that I loved math, respected my own math learning style, and got a huge kick out of helping all sorts of kids come to understand new math ideas and feel new math power.

That marked me for life, evidently. In my current pause from teaching, any time a math idea sails into my day, I grin and go with it. So, for any of you who feel math deprived, just through the holiday, or in your everyday life, I offer a few math games.

The first two aren’t really games, just reflexes.

When I tear myself out of whatever book I’m reading, I play with the page number as I walk away. 139. Hmmmm: is that prime? It might be, since none of the proper factors of 100 overlap with the factors of 39…

thinkingabout139 cropped

When someone in our family has a birthday, I figure out the prime factorization of the new age. My father recently turned 92. Let’s see: 2 x 46, or 2 x 2 x 23. Suddenly I feel, inside the 92, an 80 (4×20) and a 12 (4×3). Oooh, cool.

Last year, my daughter’s children were both prime, 3 and 7. As of a few days ago, they are both powers of 2, having turned 4 and 8 (or 2×2, and 2x2x2.) Abe is now twice as old as Julia, and that will never happen again.

What official-sounding thing can we call this? A mathematical storytelling impulse? It works for me.

But other things can work, too. I’ll never forget the day my kids and I stopped by Kate Keller’s house for a quick visit, and learned now to play Set, from watching her play it–because she refused to tell us the rules. Obsessed, we came home and made our own version out of file cards. Later on, watching my students play Set was like giving them a diagnostic test. Some kids were quicksilver zippy at Set, and slow at everything else that happened in class. Some kids were slow, as I am, but warmed up as they went along. Clues, clues. And hilarious fun: in math choice times, I had to limit the number of kids who could play Set together, because that corner would get so loud.

This is not a Set:

not set aNeither is this:

not set b

But this is:

set c croppedand this is:

set a…and this is a particularly delicious Set:

set bTo learn more, you could track down Kate Keller, my all-time-most-important math mentor, who has more fun with math than anyone else I know. She also perceives and nurtures students’ math individuality with something I can only call math compassion, a power almost magical.

Or follow this link to the Set Wikipedia entry; it’s fascinating! There are ways to play Set online now, too–a discovery that could sharply curtail my future productivity.

Finally, Lloyd’s Game. Of course, he probably called it something else. I’ve sometimes imagined Lloyd just up and quitting when one of his best whole class math games no longer worked. 1999 was a great year for this game, but the very next year, 2000, was hopeless.

In Lloyd’s Game, you have access only to the digits in the Gregorian calendar’s count for a given year. You combine those with math symbols (no quota on those) to create expressions equaling the numbers from 1 to 100. You must use all four digits in each expression, and you may not use two-digit numbers made from combinations of digits (although I remember resorting to that a few times when nothing else worked.)

Generally, in class, we used the new year’s digits to create the numbers of the days in January, catching up with a burst of activity when we came back from the holiday break or weekends, but mostly targeting each number as it came up, day by day. We used the basic operation symbols, + — × and ÷, along with parentheses, the fraction bar, the square root symbol, and the exclamation point meaning factorial. We were allowed to use a number as an exponent, so 1 to the 9th power was an excellent way to dispose of a superfluous 9.

Here are two examples, using 1989:

Lloydgame27and2 cropped

The best fun came in class, as we compared multiple ways of arriving at the same target number. Gradually, as January progressed, we watched and cheered breakthroughs for kids who had initially feared the game’s challenge.

Here are three ways of making 5, from one of the posters we hung up around the room, again from working with 1989:

LloydGame cropped

After 2000, a flop for obvious reasons, we sometimes used the year in which the largest number of kids in the class had been born. Sometimes I chose numbers relating to our themes: the year Charles Darwin was born, the year the Blackstone Canal was first opened, etc.

Try it out, alone or with some pals. You can use the birth year of your favorite politician—they’re all still creations of that wonderful (for this purpose) century past.

In any case, life is short. Go ahead and feel human. Play with math while you can!

Building Average

I’m here to confess: I’ve spent a good portion of my teaching career guiding students in freaking out the cleaning staff.

Each year, in Level 6 math, we built a model of the Average Student, statistically accurate, earnestly assembled, vaguely lifelike. We set it up in a chair toward the back of the room. Usually the students chose a book to balance on its lap. I myself sometimes entered the room, at the end of a long meeting after school, and did a double take.

Traditionally, we took a group photo of the assembled class, with the dummy. Here, for example, is an unusually small class, from the fall of 2010. (Clockwise from the top, Kelly, Ben, Seth, Anna, Lydia, and Gianna,)

average 2010 better

A few weeks post-portrait, when stray arms or eyebrows began to fall off and litter the classroom floor, we held a funeral, usually with dual caskets–since one cardboard box couldn’t hold it all. We paraded more-or-less solemnly to the dumpster, and gave heartfelt testimonials about everything Average had helped us learn–

–which was a lot. If you ask a typical adult what an average is, chances are you’ll get the series of steps followed to find the mean of a set of numbers: add up all the numbers; then divide by the number of numbers.

That’s not wrong, as directions. But what does an average really mean? What can it tell you about a situation or a set of data? What can it not tell?

MathLand­—a wonderful math curriculum no longer in print—gave Level 6 students a chance to explore the idea of ‟average” from the inside. Many years after we had shifted to another curriculum, I kept starting the year with this unit, because it was perfect from so many points of view.

Setting a goal

You could build an average kitten, or an average bookbag–but it worked really well to build an average math class student. Kids took it all more personally, and paid more attention to interesting questions: Is Average identical to any individual in the group? How does the model represent each person’s data?

MathLand provided a data sheet which included a variety of measurable attributes—such as the girth of the neck, or the length of the upper leg from the hip to the knee. The sheet also asked about attributes that had to be described in other ways—such as the color of eyes or hair.

Some questions were yes or no: Do you wear a watch most days? Some questions had been wisely left out. Average was always just Average, neither he nor she. We weren’t asked to measure around the waist, or chest, just shoulder to shoulder.

Some questions deliberately provoked discussion. How do you measure the length of the neck? From the bottom of the ear? From the hairline? The whole class had to stop and decide, together, or the data would be meaningless.

Gathering and recording data

Before we could begin collecting data, we had to choose an appropriate unit of measurement, and an appropriate degree of precision. I did specify metric units, partly because I wanted students to get some practice with decimal numbers. The kids agreed that the measurements had to be at least as precise as the nearest centimeter. Even that could result in very unrealistic hands, though; so we almost always wound up agreeing it should be to the nearest millimeter, which we recorded as a tenth of a centimeter. (Fertile fields, of course, all of this.)

Boys helped boys measure, and girls helped girls. All the data was kept anonymous—and we said that the study subjects were unreachable for clarification of messy handwriting, so the recorded data had to be both readable and reliable.

Working with data

On the other hand, the occasional inscrutable handwriting also offered a relevant opportunity, once we reached the computation stage: If you can only read the data for 11 of the 12 members of the group, what should you use to divide the total? What would happen to the mean if you divided by 12 instead of 11?

Also, once you got your mean, would it tell you anything about the huge variation in sizes of kids this age? No–only if you added information about the range, which wouldn’t actually get built into our model.

Could a very long-legged class member and a very short-legged class member cancel each other out? Yes, in effect. But in a class with several unusually long-legged people, would the mean probably be affected? Yes, again.

Meanwhile, what about the attributes described by words? For those, we found the mode, the most common answer or value, with interesting results. A math class with only 4 out of 13 blue-eyed students could wind up building a blue-eyed Average, if the rest of the kids were divided evenly among brown, green, and hazel. ‟So my brown eyes have disappeared from our Average representation?” a certain kind of kid would ask, even without being paid or prompted.

Representing data:

Ed's arm blueprint croppedAlthough they were working together, every child measured, and recorded measurements. Every child took part in finding the mean or mode for the attributes of his or her team’s assigned body part. Finally, every child drew a “blueprint.” Here’s Ed Pascoe’s blueprint for the arm and fingers.

Julia's face blueprintEach person on the team assigned to manufacture the head and facial features, for example, started out by making a basic sketch of a face, and then labeled the mouth with the mean width of the mouth, the eyes with the color of the mode for eyes, and so on. Here’s Julia Bertolet’s blueprint for the head.

Then, following the suggestion of the curriculum, but apparently against common practice in most places using MathLand, we actually built our model. We were armed:

  • with blueprints, measuring tapes and invaluable partners, for quality control;
  • with brown grocery bags for skin, crumpled newspaper for insides, Sculpey for ears and nose, and miles of masking tape to hold it all together;
  • with paper fasteners for knee and elbow joints and a meter stick taped to the back of the chair to make this character a vertebrate, able to sit up proudly;
  • with the almost invariable blue jeans and t-shirt that fulfilled those modal mandates;
  • and with endless jokes. “Where did you put our torso now?” Etc.

Being mathematicians

All this took time, it’s true. Gobs of time, all of it worthwhile. As teacher, I could observe difficulties with measuring technique, awkwardness with calculators, challenges maintaining focus even with the physical reminder of the unfinished body part. I could identify unusual ability to ask the salient questions, or to solve construction problems, or unusual gracefulness in helping a partner stay on task. The kids could figure out what to expect from, and give to, each other. I could cheer on strengths, provide the necessary re-teaching or skill-building support, and encourage insight—and kids could do all that for each other—within an atmosphere of fun.

We were having fun. We were also thinking about questions central to so many math applications: questions about reliability of data; questions about precision; and questions about whether a calculated answer fits an intuited estimate, given the range of the data. We were doing what many adult users of mathematics do: using that language to explore the world.

And of course, we were united, and found truly memorable group satisfaction, in making life more interesting for the cleaning staff. Or anyone else who wandered by.

average 2010 goofy