Dividing small by big (fraction division & pattern blocks, part 2)

Last time, I related the tale of how, with a single fifth grade arithmetic problem, Cathy Humphreys shook my confidence in my math abilities to the core and then rebuilt it again from the rubble, better, faster, stronger, because that's how she rolls.

(Do you know why 1 ÷ 2/3 = 3/2? Are you sure?

Are you?

ARE

YOU

?)

1 ÷ 2/3 is tricky because, unlike, for example, 3/2 ÷ 1/4, the divisor does not fit evenly into the dividend. But once you understand the nature of the problem--what fraction of the unit in question comprises the leftover bit?--you can probably more or less make your way through most problems where the divisor is at least smaller than the dividend.

So hold onto your pantaloons, mateys; we're about to go off the map a bit. Here there be dragons, ie, problems where we are asked to divide a SMALL fraction by a BIGGER fraction.

CRAZY TALK.

That time Cathy Humphreys taught me to divide fractions

Despite my background as a high school teacher, I've gotten deeply interested in grade 3-5 math in the past few years, particularly all the bits related to number, operations, & algebraic thinking and how they weave together to create the ramp that ultimately gives kids access to formal algebra.

But it was not always this way! As a college math major filling out applications to masters & secondary credential programs, I definitely saw myself as a high school teacher, much more interested in the complexity and rich structure of Algebra II and trigonometry and calculus than in the usual middle school topics. And I *certainly* had never gone back to closely examine my own conceptual understanding of the foundational mathematics we learn in elementary school. Who wants to teach fractions and decimals when you could be initiating kids into the wonders of trigonometric functions??

BO.

RING.

So, I got into a secondary program, started student teaching Algebra I, & learning all the magical things they teach you in Curriculum & Instruction (ie, "methods") class about how kids make sense of ideas like variables and functions and data analysis over time and what it really means to understand all these things anyway. It was mystifying and terrifying and amazing, and in addition to learning how to teach, those experiences also unlocked for me an entirely new dimension of understanding. It was exhilarating ("Who knew math could be even MORE AMAZING??) but also a bit panic-inducing ("How the HECK am I supposed to get kids to understand it THIS way?!?"). All in all, though, I was starting to feel pretty darn good about my content knowledge.

And then, one day, Cathy Humphreys came to class.

She came to teach us about fractions.

My NCSM & NCTM 2017 Highlights

Hi friends,

I spent this past week in San Antonio attending the annual conferences of the National Council of Supervisors of Mathematics and the National Council of Teachers of Mathematics as I do most years, and as always, I had a fantastic time and learned a ton.

Just a quick overview of some of the highlights:

Learning, Planning, and Teaching Together: Facilitating Job-Embedded Professional Development, Kendra Lomax and Becca Lewis (University of Washington). I feel incredibly strongly about practice-based PD, and since I also happened to know that Kendra got rave reviews from a group of K-2 teachers she'd recently been doing some practice-based/job-embedded PD with, I was excited to hear what she and her colleague had to say on the topic. We heard about how they've been exploring the idea of rehearsal as a way of teachers preparing to teach a lesson, where the rehearsing involves very focused and thoughtful real-time coaching from a teacher educator. They shared some interesting videos, and I look forward to exploring more of their work on on the TEDD (Teacher Education by Design) website.

Books: 2017 Quarter 1

As you probably already know, I've been reading a classic a month for the last two years. It started as a one-year project in 2014, but I've enjoyed it enough to keep going with it & will probably continue until it starts to feel like a chore. You can find my past reviews by clicking on the "books" tag at the end of this post, or be my friend on Goodreads. (You can also just go to the site & hunt down my review feed without being my friend, if that's more your speed.)

ICYMI, the classics I selected to read in 2017 are here.

On to the reviews!

Books: The 2017 Classics

BEHOLD! The classic novels I'll be reading in 2017:

JANUARY: The Forever War, by Joe Haldeman. "The Earth's leaders have drawn a line in the interstellar sand, despite the fact that the fierce alien enemy that they would oppose is inscrutable, unconquerable, and very far away. Private William Mandella has been propelled through space and time to fight in the distant thousand-year conflict; to perform his duties without rancor and even rise up through military ranks. He's willing to do whatever it takes to survive the ordeal and return home. But 'home' may be even more terrifying than battle, because, thanks to the time dilation caused by space travel, Mandella is aging months while the Earth he left behind is aging centuries." I like to do one true sci fi classic each year & this one got a lot of votes.

FEBRUARY: Midnight in the Garden of Good and Evil, by John Berendt. "Voodoo. Decadent socialites packing Lugars. Cotillions. With towns like Savannah, Georgia, who needs Fellini? Midnight in the Garden of Good and Evil takes two narrative strands--each worthy of its own book--and weaves them together to make a single fascinating tale. The first is author John Berendt's loving depiction of the characters and rascals that prowled Savannah in the eight years it was his home-away-from-home. Then, on May 2, 1981, the book's second story line commences, when Jim Williams, a wealthy antique dealer and Savannah's host with the most, kills his "friend" Danny Hansford. (If those quotes make you suspect something, you should.) Was it self-defense, as Williams claimed--or murder? The book sketches four separate trials, during which the dark side of this genteel party town is well and truly plumbed."

Books Update: Quarter 4

Congrats on making it through another year, you crazy diamond, you! This year I read 80 books & over 32,000 pages. We're now at the tail-end of quarter 4, so let's talk about what I've been reading over the last three months.

As you may or may not already know, I've been reading a classic a month for the last two years. It started as a one-year project in 2014, but I've enjoyed it enough to keep going with it & will probably continue until it starts to feel like a chore.

January through April

May & June

July through September

These were my last three classics of the year:

October: The Picture of Dorian Gray, by Oliver Wilde (1890, 166 pages). 4 stars. This was my spooooky Halloween read! Young, beautiful, innocent, naive Dorian Gray sits for a portrait for an artist friend, who proclaims the painting his best work. Dorian is suddenly struck by the horrible thought that he will age and lose his beauty while his portrait will remain beautiful and youthful forever. In a fit of panic he desperately prays that his and the painting's roles should be reversed, so that the painting ages while he stays young and beautiful. Not only does his wish come true, but the portrait also begins to reflect the condition of his soul. Creepiness and philosophy ensue. The writing is clever and gorgeous, and Wilde is a master of dry wit & witty repartee, plus it's less than 200 pages so pretty easy to knock out on a plane ride or similar. But, it is worth mentioning that it's still not a light read. Some parts of it are quite dense and heavily philosophical (I found myself carefully re-reading many sections because I didn't catch or process it all the first or second time around), so definitely something to save for when you are fully alert and not distracted.

Slade House, by David Mitchell

I really enjoyed this book but explaining why is complicated.

First, I should say that before Slade House, I read every other David Mitchell book except for The Thousand Autumns of Jacob de Zoet*, I believe in this order:

The Bone Clocks (2014)

Ghostwritten (1999)

Cloud Atlas (2004)

Number9Dream (2001)

Black Swan Green (2006)

Though I was struck by the gorgeous and skillful language of all five, Bone Clocks was far and away my favorite. Soon after, I read a review in the NY Times where the critic wrote that it "felt like a misfire," with a plot that "felt soft and formulaic" and read "like second-rate fantasy fiction" (of course, I vehemently disagreed!), and that the recurring characters from Mitchell's previous books seemed “less like Yoknapatawpha and more like Marvel.” Not having read any of his other books at the time, this didn't mean much to me, but I did pick up on the fact it wasn't intended as a compliment.

Over the last year, though, I've read everything except Jacob*, so now I see what he was getting at: Although the books are all self-contained, they do all seem to take place in the same universe, with the same characters or clans of characters appearing regularly (sometimes as central figures, sometimes as cameos), and taken as a whole, I could sort of see Mitchell weaving together a larger story, each book existing in its own little narrative eddy. So it was unsurprising to me to hear that Slade House was sort of a sequel to Bone Clocks.

For better or worse, that was the effect of reading Bone Clocks first. It's definitely the most explicit, sprawling, lay-it-all-out, this-is-what's-going type of story of the six, and I can see how someone might not pick up all the connections if they started with Ghostwritten and read according to publication order. I also think maybe that explicitness was what some people didn't like about Bone Clocks, having become accustomed to the oblique nature of the books that came before. (Also, I'm a huge fan of what you might call "intellectual horror" or "literary horror" or whatever, which I think Bone Clocks has a lot in common with, and I can see maybe a subset of Mitchell's usual audience not really going for that.)

So. I say all that to say that if you enjoyed Bone Clocks, you are more likely to enjoy Slade House than if you enjoyed Mitchell's other books more and Bone Clocks less. It's short, apparently an elaboration on a short story he wrote called The Right Sort. If Bone Clocks flirts with intellectual/literary horror, Slade House straight-up propositions it. The story is divided into three parts and revolves around a mysterious manor house (the eponymous Slade House), with each part following a character as they visit the house and its odd inhabitants. Not all of their fates are happy. If you've read Bone Clocks, by the second or third story you see the connection between the two books and will probably figure out sooner than other readers how it's going to end.

I can see why readers who enjoyed Cloud Atlas and Ghostwritten more might not be as big on Slade House. It's short and fairly to the point, not quite as literary, and more focused on telling a grizzly tale than on patiently and artfully developing nine-dimensional characters and abstract, highly literary themes. Still, Mitchell's skill with language and story craft in general shines through, which made it (at least for me) a satisfying, undemanding, yet highly entertaining read.

* I started Jacob in December and abandoned it after 100 pages because I just could not get into it. (Sorry, David Mitchell; six out of seven still ain't bad!)