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?

Monday, August 18, 2014

Kids Don't Need Practice Thinking Logically About Snow

I claim that this video is evidence against a common approach of teaching proof in geometry class. A weird claim, right? But I stand by it! Here's why.


Many teachers of proof and reasoning think that their students have under-developed logical reasoning skills. The way that story goes, students struggle with proof because they aren't sufficiently logical yet.

To address the deficient logical skills of their students, many teachers of geometry assign students work with non-geometric logical thinking. You'll find these exercises in many geometry texts.

The theory behind these exercises is that we can improve students' mathematical logical thinking by giving them exercises in more concrete, practical, everyday logical thinking. Since logical thinking is one skill, you can work it out in an easier context and then apply it to a more difficult, abstract context.

But the video above shows that logical reasoning is not one skill. These young kids are perfectly able to reason logically about merds and bangas, even though you can see that logical thinking come and go in the above video.

When their logical reasoning falls apart, it's for two reasons. First, there are times when the kids aren't sure whether they're supposed to use their logical reasoning or their practical knowledge to answer the question. ("This is math, not nature class.") Second, when kids/people have practical knowledge about something they tend to use it rather than reason about it. ("They do not bounce when they fall.")

Applying these lessons to math, when our kids struggle with reasoning in geometry it's likewise for one of two reasons:
  1. They don't yet really understand what's expected of them. The expectations that mathematicians have aren't always natural or easy to pick up. Kids don't always get the expectations.
  2. Kids, like adults, use their practical knowledge about the world to guide their reasoning. If kids aren't reasoning well, it's often because the practical knowledge that they're leaning on isn't strong enough yet. (It's like the van Hieles said, sort of.)
But it's not because kids can't reason logically, because young kids can reason well about merds, bangas and all other sorts of things.

  1. But clearly some people end up with super-duper logical reasoning skills that they can apply in any and all contexts. Where do those skills come from?
  2. Why do kids get better at logical reasoning about make-believe as they get older? Is it because they get socialized to the idea of reasoning about make-believe?  
  3. Take this too far, and you end up saying that if you just teach kids a bunch of facts they'll end up with the ability to write beautiful proofs. That's clearly not true. Informal reasoning about geometry is clearly important for developing your ability to reason more carefully about geometry. But why?
  4. Say that you're convinced that practice with non-mathematical logical reasoning doesn't help much with proving stuff in geometry. What could you replace that unit with that would help?

Thursday, August 14, 2014

Feedback doesn't cause learning, thinking causes learning

Arrow Feedback 
"If I had to reduce all of the research on feedback into one simple overarching idea, at least for academic subjects in school, it would be this: feedback should cause thinking. All the practical techniques discussed here work because, in one way or another, they get the students to think, rather than react emotionally to the feedback they are given." - Dylan Wiliam
Here's what I'm gaining from reading Wiliam, Butler, Kluger and Denisi. Kids don't learn anything from knowing what they did right or what they did wrong. The giving of feedback is not what directly causes learning. All the learning happens after you give feedback, when you've either sent a kid down a productive path or motivated them to reexamine their habits in a serious way.

In short: feedback that students don't act on probably doesn't do much.


  1. What does effective feedback on standard-issue quizzes and tests look like? Do you give comments on every question? The assessment as a whole?
  2. Are written and oral feedback equally effective?
  3. Are questions ("Have you considered similar triangles?") and suggestions ("Try thinking about similar triangles") equally effective when appearing in feedback?
  4. Last year I gave feedback on weekly quizzes and on some classwork. Is that enough? Is there a limit on how much feedback, practically, is useful, or is it a "the more the better" situation?

Monday, August 11, 2014

The Question-Notes-Questions Format For Self-Study

The above graph should be terrifying if you're anywhere near year five of your teaching career. Horrifying. If you're a typical teacher, the things that you learn about teaching do not make a big dent in achievement past that threshold.*

*The regular caveats apply. "Achievement" was measured using value-added models from state tests. Maybe mid-career teachers don't get better at teaching basic skills but do get better at teaching forms of creative reasoning that don't get measured on state tests? I'm skeptical, but haven't researched or thought this through enough to really defend the position. Anyway,

We teachers can be endlessly creative when it comes to creating structures to help students learn. We'll play games and run activities, we'll create rules and constraints that help nudge kids in productive directions.

What drives me nuts is that we don't apply the same creativity to our own learning. If structure helps learning and studying for kids, then it also helps adults. I think discovering and adopting strategies for self-learning about teaching is a huge key to putting a better teacher in front of my students.

The trouble with flipping through resources

A lot of my summers have been spent with this vague idea that I want to "work on curriculum." What does that mean? Usually it means that I sit with a textbook and start flipping through pages. Then any number of things might happen.

  • I notice a cool activity, I start thinking about that activity and then I go off and I start thinking about that lesson. Maybe I write it down, or maybe I just drift off and then come back and flip through the book some more.
  • I notice the order of the chapters. I start thinking about the order of chapters that I'd like. Maybe the order surprises me, maybe I start thinking that it was a mistake to really look at this textbook because I can't use it anyway because of the sequencing. Then I go back to the book.
  • Maybe there's some math that I don't understand and I work on that. Then go back to the book.
Are you like me? Does this process sound terrible? The problem is that these thoughts are disconnected and random. I end up feeling no wiser after I spend some time flipping through the book, because the things I thought about in that hour were completely isolated. I usually walk away full of ideas that disappear in an hour.

Plan for getting smarter about teaching

This summer I've been doing things differently. When I spent an hour looking at a curriculum for a new class I'm teaching, I made sure to structure my time around questions

Here's the thinking: Pushing kids to formulate questions is something that we know helps learning. So why not apply this idea to my own studying? But not only is the act of question-asking is helpful, it also helps keep me focused on productive questions when I'm looking through resources. When my mind wanders I have a decided-on point to return to, which helps me make sure that I'm thinking about questions that matter, instead of questions that just happen to be interesting at the moment.

My notes from this morning had three sections: opening questions, notes, and closing questions.

I wrote some questions down when I started. ("What does it look like when 3rd Graders don't understand place value?") I used those initial questions to guide my study of the text I was flipping through. I took some notes on things that I noticed or relevant thoughts that I had, and I also gave room to ask myself questions along the way. ("What's that thermometer question doing here?") Then, at the end of my session, I forced myself to ask some new questions that seemed worth asking. ("Why does it matter to be able to answer questions like 28 + ____ = 100? This book gives a lot of space to it.") 

I'm calling this the "Questions-Notes-Questions" structure for my studying. I made a template today so that I can plan to use the QNQ format when I need to. (link)

To be clear: I don't think that this is brilliant or innovative or anything. It's stupidly simple. But I think that being productive when you're working alone is ridiculously difficult. If I wasn't so easily distracted and flimsy in my commitments, I wouldn't need a template for my learning. But I am, so I do.

Extra Credit Questions
  1. The author claims that the strategies we use to guide student learning in class should be used to guide our own studying. Do you agree/disagree with this claim? Why?
  2. Teaching is a skill, but so is painting. Does the author's argument apply to painting? What about to other skills? 
  3. "Every study strategy is going to have some drawbacks. The question is whether the benefits outweigh the costs." Do you agree with this statement? If you do, what are the costs and drawbacks of the QNQ study strategy?

Friday, August 8, 2014

How does feedback help?

Feedback is supposed to help students learn stuff. How? By what mechanism? Why would ever feedback help?

A lot of my confusion about feedback has been about my inability to clearly answer this question. After reading a few articles and chatting with some teachers, I think that I know of two ways in which feedback helps.

Mechanism #1: You can impact what someone is thinking about.

From the  Math Mistakes website

One way that your response to someone can help them learn is by redirecting their thinking. Every comment we make to someone else has the capacity to literally change what they are paying attention to. Paying attention to something can lead someone to learn something new. 

The above bit of feedback puts little asterisks by student work that the teacher thought would be productive for a kid to spend more time thinking about. The asterisks here are functioning as an attempt to redirect the student's attention.*

*I suppose that this is open to interpretation. Maybe the asterisks are just the way that this teacher indicates that something is wrong, and maybe the kids know this. If that's the case, then this isn't just an attempt to draw attention to some piece of work but also to evaluate it, which might be something else. Anyway...

I saw an especially clear articulation of this idea in "Feedback Interventions: Toward the Understanding of a Double-Edged Sword."

"After receiving feedback, an individual is very likely to be thinking about something different from what he or she was thinking about before receiving the intervention." 

I think that asking questions in response to student thinking is a great example of trying to impact learning via redirecting someone's attention.

Mechanism #2: You can motivate a person to go after a good goal.

Also from the Math Mistakes website
It's certainly true that sometimes, when you find your performance deficient in some way, you're motivated to improve yourself. I discover that my joke didn't make anybody laugh, so I'm motivated to try and tell a funnier joke. In the process, I learn more about how to make someone laugh. Bada-bing.

It's also certainly true that sometimes this doesn't work so well. You tell somebody that you didn't find their joke funny, and they get offended or decide that they don't want to share their jokes with you anymore. You're a humorless grump. You're a critic. Or they decide that they're not funny, so they stop trying to be funny and learn nothing more about humor.

To put this into academic gobbly-gook...

This second way that feedback can help learning is all about motivation. But when does feedback help motivation and when does it hurt? How do you give feedback so that it helps? This is something that has to do with grades vs. comments, mindset, experts vs. novices, and feedback on skills vs. feedback on the task

Redirection is safer, but motivation is also powerful

I've been a heavy user of redirecting thoughts in my responses to students over the past few years. I spent less and less time comparing my students' work to learning goals because I knew of the dangers of trying to motivate them but not how to negotiate them.

I think you can get pretty far with playing with someone's attention, but I think what I really need to do is come up with a plan for negotiating the motivation-side of feedback. It can be frustrating to try to improve your performance without clear comparison, and I need to make this side of things work.

I'm just beginning this process, but I'm going to need to come up with ways to give my students goals, to give them a chance for revision, and to give comments that give kids clear and actionable ways to improve. 

A lot of this is fuzzy to me, but I think I have a clearer direction now.

Homework, Due Monday
  1. Worrying about how good you are at something can be a motivation-killer. How do you give feedback that avoids that, but that doesn't just turn into managing their attention?
  2. Are there other mechanisms through which our responses to kids directly impact their learning?
  3. When do you want to redirect someone's attention in a productive way, and when do you want to try to motivate them towards some goal?
  4. What sort of routines do I need to have to help me systematically improve my feedback-giving habits?