Saturday, September 20, 2014

Two Estimation Tasks

Here are two estimation problems.

The first is from TERC Investigations:

The second is from Andrew Stadel's Estimation180:

I have some questions about the relationship of these two problems to each other:

  1. Do these two problems help students get better at the same thing? Or do these two problems help students improve at different skills?
  2. What sort of knowledge is needed to successfully tackle the first estimation task? The second?
  3. What does the first problem seem to mean by "number sense"? What does the "second"?
  4. Can we infer why estimation is valuable to the writers of the first problem? The second?

Wednesday, September 17, 2014

How I'm Trying To Mentor Myself

Reflecting on teaching helps make teaching better, right? Blogging about teaching is a form of reflection on teaching ergo blogging makes you a better teacher.

The pic at the top of this post comes from Dan Meyer's presentation about teachers who blog or tweet and what they get out of said blogging/tweeting. I helped him a bit with the research for this talk, and the "blogging as reflection" idea came through again and again in the responses to Dan's surveys. Blogging is reflection. Reflection leads to better teaching. Better teaching? More blogging. Boom. A virtuous cycle.

I'm sure that blogging is reflection. But I don't think enough teachers grapple with the limits of blogging as reflection. And there are some pretty serious limitations.

  • It's very difficult to talk about specific incidents or students in a blog post. Readers lack the (boring) context of the classroom that could make those incidents meaningful.
  • The vast majority of people are unwilling to write publicly about their anxieties and concerns about their teaching, so the vast majority of posts are about sharing moments that worked.
  • Blogging pushes us towards reflections that can easily cross the threshold into other people's classrooms. I'm more likely to blog about "How does feedback work?" than I am about "How can I make sure that my fourth graders each have strategies for figuring out a multiplication problem by the end of the week?"
I love blogging (obviously) but lately, I've been trying something different for my reflecting. I've been calling it self-mentoring, which is good except it's a terrible name.

A brief aside: as a new teacher, my school assigned me a mentor, and it was the best learning experience about teaching that I've had so far. My mentor visited my class each week. Before she observed, we'd chat for about 30 minutes about how my week of teaching had been. She'd ask questions and push me to think more deeply about why my students were doing, thinking and saying the things that they were. Post-observation, we'd have a similar chat that was laser-focused on what had just happened in class.

I haven't worked with a mentor since that first year, but this summer I found myself wanting very badly to recapture those conversations. Truth be told, this is something that I'd been trying hard and failing to do ever since our mentorship ended. (See here here here here and here for all my public failures.) The failures were easy to come by, because the problem is fundamentally hard: how do you push your own thinking past the limits of your current thinking?

I have 30 minutes free on Wednesday afternoons, and I like the way that I've spent them over the past few weeks. I seat myself and lay out my special teaching notebook. I have a protocol for how I spend this time. Right now, it goes like this:
  1. Write about anything of interest for a moment or two.
  2. Make a list of classroom incidents that might be interesting to dig deeper into.
  3. Write down some questions.
  4. Start writing about one of the classroom incidents. I have my mentor's voice in my mind as I do this. I try to ask myself the questions she'd ask. "Why do you think that happened?" "What do you think she was worried about?" etc. 
  5. I make a little box for any significant takeaways from the analysis.
  6. I go read through the Teaching Works practices and spend a moment noting anything related to those 19 practices that might be worth working on over the next week.
I've been really happy with the way this has been going so far. It feels realer and more natural than any of my other failed experiments over the past four years. I've been coming out with a bunch of takeaways that would make for terrible blog posts: "I sometimes walk away from a kid before I understand their thinking, and this makes it hard to decide who to choose to share in discussions" and things of that sort.

It's a lovely little ritual. I hope that I stick with it, but I think that I will because it's a lot of fun and it's been helping me make improvements. I'm hopeful that I've landed on something that works for me.

Sunday, September 14, 2014

Do kids need help learning how to ask interesting questions? (Continued)

"Let's call out the elephant in the room: we're in math class. Chances are extremely high that mathematical questions are preferred. I think this should be discussed with students AND modeled by the teacher." (-Andrew)
"I think kids think it is reasonable to focus on math in math class, especially if they believe us when we tell them it's useful and lets us find out cool things." (-Julie)
"So when we ask for questions, we honor all the questions. We're grateful for all the questions. Then we model how mathematicians think, how mathematicians ask questions, and we ask our OWN questions that we know lead towards productive mathematical goals." (-Dan)
I was grateful for the comments on this earlier post. It seems to me that while many of you agree that mathematical questioning ought to be taught, we're caught between two different teaching strategies.

Strategy 1: Implicit Instruction

Honor every question or curiosity. Get excited by what the kids are excited about. But then always pick up on the questions that are mathematically productive. Say things like "We'll try to answer as many of these as we can." and "I love all these questions, but today we're going to focus on..." Say these on many days.

Show that you're excited and curious about your own question. Show that you have questions that animate and excite you too, and they just happen to take us to interesting mathematical places.

The kids will pick up on this, if not at first then with time. They'll want their questions to lead to the interesting mathematical discussions. If only mathematically productive questions lead to prime-time, they'll learn to ask mathematically productive questions. They'll be standing at the edge of a culture, peering in and trying to fit-in. It'll take time, but it'll stick.

Strategy 2: Implicit Instruction + Explicit Instruction

Help kids see things that they'd struggle to notice. Speed along their learning process by pointing out aspects of questions that make them mathematically purposeful. Say things like "What interesting mathematical questions could we ask?" or "I love questions that aren't just a matter of opinion."

Honor questions, yes, but also use praise, questioning and advice to help guide students towards productive lines of questioning: "I love 'How many?' questions."

Asking mathematically productive questions is hard. If we rely on implicit instruction then maybe some of our students will learn to ask great questions, but it's unlikely that all will. And if we want equitable outcomes, shouldn't we care about helping all students get there?

Explicit instruction can be done well and need not involve being dismissive of a student's natural curiosity. We use explicit instruction to teach all sorts of concepts, skills, strategies and practices. Why not use it to help students ask productive questions as well?


  1. When is implicit instruction preferred over a mix of implicit and explicit strategies? Why?
  2. Is "modeling" a form of "implicit instruction"? Are they synonyms? Are there are other forms of implicit instruction?
  3. The concern with explicit instruction seems to be that any explicit instruction on questioning would necessarily show that natural curiosity isn't valued by the teacher. But implicit teaching techniques aim to alter student questioning, anyway. Is the difference whether students notice that they're being guided? If not, is there some other difference?
  4. Is there necessarily a trade-off between equity and discovery in teaching?

Tuesday, September 9, 2014

Written Feedback: Four Cases

Four quick summaries of the problems that students worked on, and the feedback that I gave each group the next day. A snapshot of the group's product before and after feedback.

Case 1: Handshakes and Diagonals
  1. How many handshakes would it take for us to all shake hands?
  2. How many diagonals does a square have? A pentagon? A hexagon? Find a relationship between the number of sides and the number of diagonals.

Case 2: Grids and Squares

How many squares can you find on this square grid? How about on this array of dots?

Case 3: Nets of Cubes

How many nets does a cube have?

Case 4: Cross-sections of Cubes

Here's a list of shapes. Can these shapes be the cross-section of a cube?

  1. Were all of these pieces of feedback equally effective, based on the whiteboards alone?
  2. Each of these pieces of feedback were printed out and handed to a group. Would oral feedback have been better here?
  3. Each of these pieces of feedback ended with a question. Would a direct suggestion have been more effective here?

Monday, August 25, 2014

Do kids need help learning how to ask interesting questions?

Lifted from Alan Schoenfeld's Learning To Think Mathematically:

As teachers, we want to value every question that comes out of a kid's mouth. But not all questions are equally valued in math -- asking mathematically productive questions is a skill that successful math students learn. 

Take this enormous badminton puck or whatever you call it:

You ask most students: "What do you wonder when looking at this picture?" Tell me if I'm wrong here, but I think that they'd respond like this:
  • What is that?
  • What's that thing in the back?
  • Why did someone put it there?
  • Is that a sculpture?
  • What's it made out of?
  • Where is this?

And so on. Compare that with the list of questions that a bunch of teachers came up with:
  • How big would the racket need to be?
  • How big would the person who could hit this thing be?
  • What time is it?
With time and practice, the students will more closely resemble their teachers. They'll notice what questions get picked up in class for lengthy discussions. They'll compare their questions to the questions their teacher asks. Slowly, they'll get a nose for the sorts of questions that animate mathematicians.

Asking a good question in math isn't natural. It's something that people learn how to do.

Which leaves me with a big question: how important is it to play to students' natural curiosity? Should we follow Annie Fetter's lead and ask students what they wonder? Should we follow Dan and ask kids what the first question that pops into their head is?

On one hand, the case for natural curiosity seems strong. If we care about what engages kids, it's important to know what kids are interested in. If we care about assessment, we shouldn't want kids to hold back. And if we care about including everyone, there should be a low barrier to participation.

On the other hand, the ability to ask an interesting mathematical question is something that is learned, and it is important. And if something is important and capable of being learned, shouldn't we teach it?

  1. Should we also be worried about teaching productive question-asking? Can we teach it effectively if we always play for our students' natural curiosity? 
  2. From Max Ray's Powerful Problem Solving: "Noticing and wondering is something students get better at over time." How do kids get better at being naturally curious? Are we changing what they're naturally curious about?
  3. Are there times when we want kids to be naturally curious and times when we want them to be unnaturally curious? When would those times be? 

The Last Post I'll Write On SBG and Feedback, Probably

That picture makes me a little icky inside. As of two years ago, this was the feedback that I was giving my kids. They had weekly quizzes, and I was using Standards-Based Grading for reassessments and all that jazz. They were finishing their quizzes and then marking up their own quizzes against an answer key. Then I would go through it again and mark up their rubric levels for each skill at the bottom of their quiz.

This was not good feedback. Good feedback would have involved giving each kid some way to improve or advance their work on the quiz.

Further, even if I had been giving good feedback in addition to these SBG scores at the bottom of the quiz, it probably would have been a waste. Kids would likely just skip over my comments and go straight to the SBG scores on the back.

SBG might be great for all sorts of reasons, but the way I was doing it didn't really allow for much in the way of good feedback. I don't want to tell you that you shouldn't do SBG, but I don't think that we should compromise on feedback. Unlike what some people will tell you, SBG is not good feedback.

(This ends a series of posts starting back in January 2012.)

Thursday, August 21, 2014

"Draw a Picture" is Too Darn Abstract For Kids

Strategies are a good thing. They help you solve problems, they can get you unstuck, but when I suggest them to students I'm often met with a blank stare: "Uhh. How do you find a pattern here?" Or, I'll ask a kid who is stumped, "Can you draw a picture for this problem?" and they'll nod and I'll walk away for five minutes and when I come back they've drawn a bakery.

Kids don't get what these problem solving strategies mean. And I think that's because we teachers often don't realize how abstract problem solving strategies like "draw a picture" or "work backwards" are. I'd like to speculate that these strategies are just as abstract for our kids as variables, equations or other high-level representations like graphs.

But what does it mean for a strategy to be abstract, and what does it mean to abstract a strategy? Dan has really thought through what it would mean for "to abstract" to mean "to remove irrelevant details," and for an abstraction to be the result of this process. But I think that there's a flipside of that, which is that when we're abstracting we're bundling. That's important enough to my argument I'm going to write that in really big letters before attempting to defend it. One second,

Abstracting as Bundling

...and we're back! I think that a lot of what it means to abstract something is to bundle it together with a bunch of things. So, for example, you start with Bessie.

But then you add Judith, Ann and Cleopatra to the mix.

You notice some things that are similar about these four fine gals. You notice some things that are different. You decide that they're similar enough that they deserve a name, so you create a category called "cow." "Cow" is all about the important things Bessie and Friends have in common, and (as Dan says) ignoring the parts you don't care about.

One way of looking at this process is with the metaphor* of the "ladder of abstraction." I like the idea of a ladder, but it makes me sad that Ann, Cleopatra and Judith don't get to be on the ladder since they were so important to this whole deal. So I made a tree diagram that has room for them.

* More like MEAT-aphor amiright?

My contention is that this depiction of abstraction is just as true for strategies and skills (e.g. “draw a picture”) as it is for concepts (e.g. “cow”). A strategy such as "work backwards" sits at the very top of a very tall tree of abstractions. There isn't just one thing called "work backwards." Well, there is, but it's a family of (families of!) strategies that have been bundled together under this one name.

This is a point that has been made by Alan Schoenfeld:
“A major realization was that Polya’s descriptions of the strategies were too broad: “Try to solve an easier related problem” sounds like a sensible strategy, for example, but it turns out that depending on the original problem, there are at least a dozen different ways to create easier related problems. Each of these is a strategy in itself.”
It’s great to have a top-level strategy like “solve an easier, related problem” if you have lots of experience with all the strategies that are under it. But if you're don’t, then when someone tells you “Try solving an easier problem!” it's like pointing at Bessie telling a kid “Do you see that wealth out there, chomping grass in the field?”

We’re jumping to an abstraction, and the communication fails.

What is a teacher to do?

When it comes to teaching the big math skills and strategies, I think that we have to move up and down this ladder of abstraction. In particular, we want to help kids organize their learning more effectively by helping students bundle their strategies into more general, more widely-applicable packages.

Consider this kid, who is getting pretty good at drawing a picture for certain kinds of multiplication problems.

In fact, she can do this for addition problems and fraction problems too. She knows how to draw pictures to help her solve problems in lots of different specific situations. This is where the teacher's work begins.

One way that we can help this student is by encouraging her to see these three specific drawing tricks that she already knows about as a strategy. We're asking her to do some bundling. Instead of Wilcox, Ann, Judith and Cleopatra, we're asking her to bundle "draw a picture to help solve a multiplication problem," "draw a picture to help solve a fraction problem" and "draw a picture to help solve an addition problem," into a single category.

Now, say this kid is stuck on a subtraction word problem and you advise her, "Try drawing a picture!" There's a decent shot, I'd argue, that this kid will be able to pick a representation for this problem. If not, that's cool, push her to think about how she'd represent this if it were an addition problem, or a division problem. Draw connections that will help her bundle her strategies and ascend higher and higher up the ladder of strategy-abstraction.

Eventually, this kid will become pro at drawing pictures to help solve arithmetic problems. Does that mean that she'll be able to draw a picture to help her solve a linear equation? Of course not! Even if she understood variables and equations, do we expect her to easily intuit that a scale is a helpful way to visually represent a linear equation?

No. She needs to learn some powerful new pictures to draw. But once she does learn this, we should encourage her to connect this to the other pictures she knows are useful to draw for algebra problems. And throughout this process we should encourage her to connect these sorts of pictures to the representations she draws for other sorts of problems. We need to encourage her to see how all these strategies are very similar, and truly belong under one bundle.

Is this enough to give our student a general-purpose problem solving strategy yet? Probably not! This process of bundling and packaging is going to continue throughout her mathematical career. At some point she'll have seen a lot of math problems where visualizations can help and, when facing something new, she'll ask herself, "Hey, I wonder if I can draw a picture here."

But the point is that there's a long development to this point, and I don't think that it's enough to just ask kids, “Have you tried drawing a picture?”

  1. We need to teach kids lots of more specific strategies, but should that happen instead of teaching top-level strategies, or while teaching the most abstract strategies? What's most effective?
  2. Do we just want kids to notice "Hey! A picture I drew helped me solve all these different sorts of arithmetic problems." Or do we want them to notice similarities and connections within the types of pictures that ended up helping them?
  3. Where does this leave our posters of problem solving skills? Do these things help, in the end?
  4. What would it take to fully articulate the landscape of "draw a picture" or "think logically" or "work backwards" that it takes for a student to reach a serious level of expertise? Any grad students out there wanna pitch in on this?