Thursday, January 30, 2014

Yarmulke Tales, ctd.

[We were adding 1/3 and 5/6.]

"I have a question. It's only a little bit off topic, but it's related."

"Yeah, sure."

"Do kosher people eat Swiss cheese?"

#yarmulketales #mathchat 

Tuesday, January 21, 2014

Quick Curriculum Note: Introducing Polar Coordinates

I introduced polar coordinates today by blindfolding one student at a time, sending her out of the room, and drawing a point on the blackboard. While the student was out, I picked someone to guide the blindfolded kid to the point. They were allowed one sentence. I brought the kid in, put her finger at the center of the board, and then she got her instructions.

Roughly speaking, there were two types of directions:
  1. "Take 5 steps to the right, and then move your finger up a foot."
  2. "Move your finger diagonally from the center for a foot."
There's some technique involved in picking points that will provoke the sorts of responses you'll want. To get talk of "diagonals," keep your point near the center of the board so that no steps are really involved. (Why? It's harder to measure inches than steps, so kids go for directions and angles.)

Sunday, January 19, 2014

What Proof With Little Kids Looks Like

Way back in October, Kate asked: What does proof with little kids looks like?

Here's an answer from some of my fourth graders:


Here's the lesson:

I walked in to class and told the kids that I had been walking down the street and found scraps of kids' subtraction work. I smiled. They asked if I was lying, and I lied and said that I wasn't. Then I said, also I found the original problem. They thought this shtick was pretty ridiculous. Anyway, I gave them this:

They did this:


Then we went through each "kid"'s work and predicted what the rest of their work was going to look like. Then I revealed it, and we tried to figure out what the kid did.


Ben Blum-Smith wants us to name our problem-writing techniques, so I'll file this one under "hide the math."

[The source for those pictures was this really great NCTM article, by the way.]

Thursday, January 16, 2014

A Chinese Proof of a Vertical Angles Theorem

Exercise: Translate this proof.

[Source: Jacobs' Geometry text. It's a pretty typical Geometry book, but this is just a great idea for a problem.]

Monday, January 13, 2014

The 10K Chart

I don't take any credit for any of this. The Investigations curriculum took me 99% of the way there with their 10,000 chart.

It's just a great idea for an activity. But the questions that they suggest seemed sort of lame to me.

Their suggested lesson sequence is:
  1. Show the chart
  2. Ask kids: How many squares are on the chart?
  3. Ask kids: Where would 50 be? 100? What other landmarks should we put on the chart? 
  4. Finally: Where would 7,396 be?
If there's anything that I've learned from Dan Meyer, it's that questions 4 and 3 should be swapped. That's what we did, and it went great.

[The post-its mark the kids initial guesses as to where 7,396 would be.]

Prep time was about 45 minutes, since I had to make a few files for this activity. It took me 30 minutes to paste the 10k chart together, but you're probably better with scissors and tape than I am. Files are here, if you'd like 'em.

Monday, January 6, 2014

The Unhelpful Distinction Between Pedagogy and Content

Geoff writes:
I’ve been truly enjoying Dan’s blog post series (from last year!) on the “Fake World.” I’d highly recommend you go read those posts if you haven’t already. It deftly exposes the fallacy of authenticity-as-engagement. I would like to offer a defense (three, really) of applied mathematical tasks.
He offers three arguments. Here's the third:
Also, by dismissing the “real-world” as a lever to engagement, you’re giving teachers a kind of out. I’ve had conversations with sanctimonious math teachers and district instructional coaches that cite Paul Lockhart as a reason to keep doing what they’re doing. I’ve read Lockart. I love Lockhart. But his books aren’t about instructional practice. While much of “Measurement”, say, can and should be handed over to students to explore, it’s frustrating to kids who have only experience math in the abstract.
Of the three, this is the only of his argument that draws blood for me. If you love math, you end up loving the fake world, and it's tempting to think that others will share that love. But the fake world is tough for kids. It's very, very different from the world in which they inhabit.

But so is the real world, no? Dan's done a great job making that case. So if the fake world isn't real for kids, and the real world isn't real for kids, then that means that...

...the world is fake for kids.

And that sounds closer to the truth for me. The world, as experienced by a mathematician, is very different than it is for your average civilian. You learn a bunch of math, and you start seeing things differently. You talk differently. You find yourself asking all these questions that nobody else asks.

And that's because no content is a "natural" context for mathematics. Or rather, no content is inherently engaging, and the world needs a different language for talking about student interest than engagement with content.

We have to change the conversation. Content isn't engaging, not all on its own. Pedagogy partnered with dumb content can only take you so far. What we need is a way to talk about partnering great content with effective pedagogy, what I'm calling "teaching" until someone comes up with a better term for it.

(Heavily influenced by this post from Larry Cuban.)

Sunday, January 5, 2014

My Education-Irrelevant Internet

I'll be a man about this and just admit that I'm sort of imagining that you'll read this post and share your favorite internet things too.


  1. Ta-Nehisi Coates, writing for the Atlantic. Recently he's been writing about postwar Europe, but most often he writes about blackness and race, and this is one of the most amazing things I've read in a while. Most important? He's got the best comments on the internet, no contest.
  2. So you don't like comic books? You probably won't be able to keep up with comic writer Brian Michael Bendis, then, since half of what he posts are images from comics. That other half, though, is essentially him talking with fans about his tastes and writing process. Today he answered a question about humor that really knocked me out.  
  3. Joe Hill's twitter feed hits so many of my interests. First, he's a really awesome horror writer, and I always like hearing about his process. Like any good writer, he reads voraciously, and he's kind enough to share it all. He's often thinking about how to limit his tech use. I remember another great thing he wrote -- sorry, no link, it was way back when -- where he said that in his house he and his sons have a "no tech room," where no devices are ever allowed. Not so useful for those of us slumming it up in our first apartments, but still...
  4. When it comes to politics, I don't know much, and I try not to form opinions without earning the right to them. Anyway, I regularly find The American Conservative's blog interesting. Last week there was a post about how a Moroccan private college is using internships to send relatively well-off students into lower class social contexts. (Was that the right place to link to the article? I never know...) The blog's kinda highfalutin, but interesting.
  5. I'm actually not a huge sports fan these days. That doesn't get in the way of enjoying Joe Posnanski's long examinations of the Baseball Hall of Fame, or his frequent use of statistics to find stories worth telling. (BTW, he's one of the good guys.) He writes in a couple of places, but go check him out here.
This is going to be fun. The blog may be dead, but who cares? Drop your favorites in the comments, or (better yet) blog about it on your own place.

Help A Rookie Elementary Math Teacher Teach Addition And Large Numbers

A teaching rule I have for myself is: Always try to follow the units of a solid curriculum, even when I'm completely ignoring their individual lessons.

It's a good rule. For one, it imposes a structure on the year, a net for me to play tennis with. It also means that I get lots of the benefits of a curriculum without losing any flexibility. By sticking to the units I make sure that, on a macro level, the material that we're learning is connected and that topics are reviewed and previewed in a sensible fashion. (An added bonus is that if I use the big-level sequencing of a curriculum, and I can use a lot of their homework materials, cutting down my workload.)

(Another teaching rule: Homework is sometimes necessary, but its impact on my life should be minimal.)

Anyway: I'm prepping for the next few weeks of 4th Grade right now, and my usual rules and tricks are faltering, and I could use some help.

The TERC Investigations curriculum has, as its next unit, a series of lessons about place value, addition and subtraction and numbers up to 10,000.

The problem is that I'm pretty sure that this stuff is all going to be too easy for my kids. So, just skip the entire unit. Well, for a couple of reasons I try to avoid that. First, maybe my kids are 85% of the way to where they need to be. That's close enough that this unit might be too easy, but not a waste of their times. Second, I'm sure that I have some students who would super-duper benefit from this unit.

So, just make the unit harder. How? By bumping up the numbers that we're dealing with by several orders of magnitude, I guess.

But the other problem is that I don't really have any mathematically juicy lessons, either for addition or large numbers. After a bit of thinking, I was able to come up with some preliminary ideas:
  • Make a book with 100 numbers on each page. What page would 45,321 be on? What about 12,000,345?
  • If we could deal with some juicy visual contexts that involve huge quantities, then we could end up with contexts for lots of big-number math.
  • Pick random big numbers; pick random kids; ask them to add random numbers in their brains, share strategies; rinse; repeat.
  • More? Help?

I'm also still not really sure where my kids struggle. Like, they definitely sometimes get freaked out by large numbers, but what sorts of mistakes or misconceptions are they going to have with this stuff? What is it going to look like for a kid to mess up with large numbers or big addition?

Any help that you guys could provide here would be helpful. I'll hang out in the comments.

Friday, January 3, 2014

Books Worth Reading: "Woman's 'True' Profession"

In America, teaching is predominantly a job for white females.

It's also a relatively low-paying job, even compared to other countries.

I recently picked up "Woman's 'True' Profession" in hopes of understanding a bit more about how we got here. Already it's fascinating. 

Back in the day teachers were mostly men:
The teacher of the village school was usually a man, as were the teachers in urban areas. A student of the ministry or at college to learn a profession, he taught not for love but to earn money during his long winter vacation. Farther from the city, the teacher was often a college dropout or a fellow with some handicap that ill-suited him for farm life.
Wondering where "Those who can't do, teach" comes from?
Said one forthright commentator looking back in 1890, teaching was a "half-house for those bound for the learned professions, and a hospital for the weak-minded of those who have already entered them."
But all this changed with the creation of our modern school system and industrialization:
In Massachusetts where feminization happened earliest, between 1830 and 1880 a quarter of all native-born women who worked outside of the home were at one time teachers. Their tenure was on average two years.
There are a whole complex tangle of causes for this movement that are identified by scholarship, but I'm still getting the hang of them all. One thing that's especially cool is that in the 19th century everyone was saying that teachers ought to be women, because of their motherly habits.
Catherine Beecher, the early and eloquent spokesperson for woman's profession, used the doctrine of separate spheres to do 'ideological work.' [...] Women were more suited than men to the work of human development, she argued, because they were more "benevolent" more willing to "make sacrifices of personal enjoyment."
We're pretty much stuck with these two images these days. Teachers are still either seen as hapless men or as caring and supportive surrogate parents.

It's starting to seem to me that gender is a helpful lens for understanding the current education reform movement, but all of that's for another day. I'll be sure to share more cool stuff as it comes along in my reading.

Wednesday, January 1, 2014

My 2013 Was Better Than Your's

It's the New Year, which means that I have the time but no motivation to write a substantive blog post. This calls for some reflection!

This past year was an amazing year for my blog. I saw a huge growth in visits to this site:

Blogger is telling me that I had about 11 billion viewers last month. If you stacked all of those people atop one another, they would be 11 billion people tall, which I think is pretty astounding.

One of my goals for next year is to double my readership. I intend to do this by continuing to focus on some of the most important (and controversial!) topics in education today. These include such important issues as
  • Exponents
  • Some cool math mistakes that I've found
  • Lists of things that I found interesting as I was walking to the subway
  • Disagreements on twitter
...and more!

Finally, I want you to know that I am a humbled by all of the support I've gotten in the past year. Really. Like, you wouldn't believe the support that I've gotten. It's freaking enormous. And I'm humbled by it all.

Wishing you and me (but mostly me!) a great 2014!

Update: On a reread, this comes off as more sarcastic and less fun than intended. All I was aiming for a playful poke.