## Tuesday, December 30, 2014

### Year In Review

Here's what I wrote about in 2014:

Making Sense of Complex Numbers - I wrote a series of posts developing an argument against the idea that complex numbers were introduced for "whimsical" purposes. (this, then this) Teaching and reading about elementary school math helped push me to the conclusion that rotations might be the best way into imaginary numbers. At the highest level of sophistication, though, complex numbers are nothing more than points. So: start with rotations, end up with points.

Feedback and Revision - Before this year I was pretty pessimistic about feedback. If not pessimistic, then confused. I started making progress when I realized some potential drawbacks of immediate feedback. The next step was understanding how feedback could come in the form of questions and that feedback didn't even need to be individualized. (Though it could.) It was hard to square these conclusions with SBG-as-feedback, and I spent some time worrying about that. But I also ended up worrying about how we talk about feedback, and this lead to my series of posts about feedback and revision.

Exponents as Numbers - Past research made me very familiar with the sorts of mistakes that kids make with exponents, but I didn't really have a prescription. I still don't, but a few posts this year brought me closer. I argued -- and I don't know if I still agree with this -- that a sophisticated understanding of exponents is closer to seeing exponents as a number and not as an operation and certainly not as an abbreviation. How do we help students see the sorts of numbers that exponents represent? I think geometric series are key. I argued that we could use this to give kids a sense of what a "power" is. I then wrote about two lessons that show how I tried this in class last year. (here and here)

Teaching Proof - Much of my writing about proof this year has a focused thesis: proof is hard because geometric reasoning is hard, not because logic is hard. In fact, to the extent that logical reasoning can be called an independent skill, children (even young children) already have it. To show this I asked parents to try an experiment with their kids and the results were clear. Rebecca's children can reason logically, so can these kids, and so can your's. This goes against a popular teacher account of why students struggle with proof, and it has implications for instruction. We need to spend less time teaching "everyday" logic and more time scaffolding geometric proof with the full range of proof activities. In general, I came to think that we need to get more specific about the proof-knowledge our kids are missing.

Researchers and Teachers - Some of the most fun I had this year was reading and writing about From The Ivory Tower to the Schoolhouse with Raymond. (Thanks, Raymond!) The book is all about the ways university research does and doesn't make its influence felt in classrooms, and our posts dug into these ideas. The trouble is that good ideas aren't always popular ones, and a lot has to do with the popularizer and the message. These sorts of concerns popped up when I wrote about feedback and generally caused me to be anxious about my own career.

***

This list is disparate. Does anything unify these concerns?

Besides for a pain-in-the-ass contrarian streak (but isn't every argument contrarian?) I think that my writing this year struggled mightily with the theory/practice divide. Teachers that I know (myself included) tend to seek activities and easily usable answers and resources. But on the topics that I've thought the most about -- proof, exponents, feedback, complex numbers -- I see the existing answers as inadequate. Teachers aren't theorists, though, and the way that we communicate is through easily usable activities and resources. (That is, sharing resources is teacher discourse.)

What will it be: essays or resources? This next year I'd like to do a better job with both.

## Saturday, December 20, 2014

### In Conclusion (Post 10 of 10)

In conclusion, teaching has a writing problem.

Part of the problem, I've been arguing, is that in education we talk about the wrong things. We talk about concepts that don't get at the heart of classroom teaching -- concepts like feedback -- and so we ask questions that are impossible to answer. As a consequence, these questions -- like how do you give effective feedback? -- are very difficult to respond to. They require the ability to split hairs and to generalize. Academic researchers and consultants are very well-suited to these tasks. Teachers aren't.

And so almost all the writing about teaching comes from researchers or consultants, people who are no longer k-12 classroom teachers. This would be a fine state of affairs if it worked. But writing about teaching rarely does.

My favorite writer about feedback is Dylan Wiliam. His work is as smart as Edutopia's is silly and reductive. Here's a picture that makes him look like a super-villain:

I heartily recommend Embedded Formative Assessment. There's a great passage in there where Wiliam realizes how difficult it is to communicate about feedback.
In 1998, when Paul Black and I published "Inside the Black Box," we recommended that feedback during learning be in the form of comments rather than grades, and many teachers took this to heart. Unfortunately, in many cases, the feedback was not particularly helpful. Typically, the feedback would focus on what was deficient about the work submitted, which the students were not able to resubmit, rather than on what to do to improve their future learning.
Got it! Focus on what the kid can do to improve, right?
I remember talking to a middle school student who was looking at the feedback his teacher had given him on a science assignment. The teacher had written, "You need to be more systematic in planning your scientific inquiries." I asked the student what that meant to him, and he said, "I don't know. If I knew how to be more systematical, I would have been more systematic the first time."
So Wiliam needs a way to say exactly how feedback can tell a kid how to improve. What does he land on?
To be effective, feedback needs to direct attention to what's next rather than focusing on how well or badly the student did on the work.
Feedback should cause thinking.
If, however, we embrace the idea of feedback as a recipe for future action, then it is easy to see how to make feedback work constructively.
It seems to me that Mr. "Be More Systematic" probably thought that his feedback focused on what's next, caused thinking, and gave a recipe for future action. Wiliam's metaphors and slogans are not enough.

---

It's not Wiliam's fault. He's trying to tackle an incredibly broad subject. (The chapter is titled Feedback That Moves Learning Forward. All of it??)

Here's an embarrassing belief that I have: classroom teachers have a chance to make things better. Nobody in education is better equipped to find the most helpful way to think or write about classroom teaching than classroom teachers. We have the stories. We have the teachers -- novice and experienced -- that we talk to daily. We know what our colleagues find helpful. We know our kids.

When we write about teaching, we have to play to our advantage. We have experience, stories and students to work with. What we lack is precisely what researchers and consultants have -- wide-ranging perspective and time. I think a lot of the silliness we see in education writing comes from writers trying to hit the scope and abstraction of university research. (It's like writing an op-ed in binary.)

There is so much fantastic story-telling from teachers that I read, but just telling stories isn't going to be enough. If we want to contribute to our common knowledge about teaching we need to try to do something with our stories. To analyze them, to collect them, to point out patterns and to generalize from them. That's one of the challenges that teachers writing about teaching face.

Maybe. I think.

Questions

1. I've argued that revision is a central concept in teaching. No way this covers the entire (but wildly abstract) formative assessment landscape. What other core concepts could we break formative assessment/feedback in to?
2. It's easy to help other teachers improve their repertoire of activities or teaching moves. (Just tell us, and we add it to the mental list.) But can we also share our decision-making, how we navigate through our repertoire?
3. Is there an intellectually respectable way for teachers to write about teaching that other teachers will want to read?
4. Is there an intellectually respectable way for classroom teachers to write about teaching that university researchers will want to read?
This is the last post in a series about feedback. To read the rest of the series, click here.

## Tuesday, December 16, 2014

### How To Write About Teaching... (Post 9 of 10)

...question mark? This is really an open question for me. I am far less confident about how to write about teaching than I am about how to teach.

Here's what we know: teachers don't like educational research.

Another thing we know: there is a lot of educational research about feedback.

Look: if you're reading this then you're probably a math teacher. And we've already established that math teachers don't really care for educational research. You might have a faint interest in what the research says, but in all likelihood you don't find that research compelling in any way.

Q: What does compel a teacher, then?

A: Twitter.
Where is the middle ground here? Social media doesn't offer much argument, but academic research is toxic for teachers. Is there any intellectually respectable way to talk about teaching that teachers will care about?

Like I said, to me it's really an open question. I don't know how to do it. I'm struggling here.

My recent writing about feedback is an attempt to find a place to land. Here are the (loose) guidelines I had for this series of posts on feedback:
• Talk about a kid, not "the kids." Whenever possible, I tried to focus in on a specific student. That helped me give enough context about the kid's specific needs (social, cognitive, motivation, etc.) that you could understand my situation better.
• Talk about decisions instead of what happened. I tried to avoid the "Here's what I used to do...here's what I did instead...it worked!" cliche of teacher-writing. It's hard to know what's really happening in someone else's classroom. I can't help you see that my feedback is working, and you shouldn't just take my word that it is. Instead, I tried to argue for the way that I was thinking about teaching.
• "Enough Context" = The math goals + kid's thinking + kid's social status. It was important to me that I give you enough detail so that you could be capable of skepticism. I found that three things often felt important to offer: what I was trying to do, what the kid thinks about the math, and how he/she interacts with math and her peers.
Here's what we know: teachers don't like reading research, but we have no good substitute for it. In a world like ours, the only sensible thing to do seems to be to read research but write from experience. But experience is famously unreliable. We need to find some way to raise the bar on writing from experience.

But I feel very unsure about this whole thing, so here are some questions.
1. The nutty thing is that sometimes teachers -- even the very teachers who poopoo research! -- cite or rely heavily on the research world. "Research shows that white boards work," or "Science shows that timed tests are bad for kids." I mean, what's the deal here? (See Schenider, to start?)
2. Is blogging the solution?
3. Forget research for a second. What do you like/hate about non-academic writing about teaching?
4. The test for teachers often seems to be usability. "It has lots of practical, simple suggestions that I can use in class the next day." Is this what's missing from writing about teaching? Maybe a mix of instantly usable ideas with longer-form arguments could help?
This is the ninth post in a series on feedback. To read the rest of the posts click here.

## Sunday, December 14, 2014

### Does Feedback Need to Be Used In Class? (Post 8 of 10)

I chat with people about feedback and revision online. Here's what I notice.

• Everybody agrees that feedback is important.
• Most teachers agree that revision is important.
• Very few teachers are giving students a chance to revise their classwork in class.

What are teachers asking students to do with their feedback? They either loosen the "in class" or the "revise their classwork" requirement of that revise their classwork in class formula:
• Kids are given the option to resubmit their work, but they're expected to do the revision on their own, outside of class.
• Kids are given feedback and they have future chances to improve their performance on a different, but related problem. (I'm really thinking of the SBG crowd, here.)

This post is here to report that I haven't had much success with either of these common practices. If I want every student to work on something, I find that I need to ask them to work on it in class. And I find that if I just give feedback without giving students a chance to use that feedback (more-or-less) immediately, that feedback tends to be just another thing that I've said instead of something that sticks.

Maybe things are different for your kids? I really have no clue, and would love to know.

In sum, here are some questions that I have.

Questions
1. Do your kids remember feedback that they don't immediately use in some further classwork?
2. Are there ways to ensure that your students are thinking about the feedback that you give without asking them to use that feedback on a problem?
3. What's the theory behind how feedback helps students do better on future tasks, if they aren't using that feedback to practice?
4. Do your kids do quality work outside of class? I've never really been able to coax quality out-of-school work from children but kids write essays so it must be possible.
This is the eighth post in a series on feedback. To read the rest of the posts click here.

## Wednesday, December 3, 2014

### Frequently Asked Questions About Feedback (Post 7 of 10)

Q: What does effective feedback look like?

A: It's really best not to worry too much about this. Effective feedback looks wildly different in different situations.

Q: What does effective feedback look like?

A: Seriously, don't sweat this.

Q: What does effective feedback look like?

A: Look I GET IT. You want to know how to use feedback to move your students forward. Me too. But all of the guidelines and slogans you hear flying around about feedback are just plain silly. "The sooner the better"? "Personalized feedback is always better"? These guidelines are wrong, and it's sort of a crazy way to talk about teaching.

If you knew why you were giving feedback, you wouldn't need to ask what effective feedback looks like. We need to shift the conversation in two ways: (1) Away from "what good feedback looks like" to "how do we make good decisions about feedback" and (2) away from "giving feedback" to more specific teaching moves, such as asking for revisions.

We need to focus on decision-making instead of just the product, but feedback isn't specific enough to really gain clarity from thinking through. That's why we should move to thinking about revision.

Q: Why are you yelling, can you please stop yelling?

A: In fact, lots of "bad feedback" is frequently helpful! I mean, if you listen to the way some educators talk it's like there's this golden ideal of narrative, written comments that look a lot like the sorts of notes you're supposed to get on a research paper or a short story. And, sure, that sort of feedback can sometimes be helpful. But when? Why? There's no theory, so no wonder that we end up focusing on what effective feedback looks like.

But "revision" is knowable. We can study it. It encompasses feedback, because feedback is what it's going to take to help students improve their work. At the same time, it's easy enough to pin down "asking for revisions" that we can actually say things about it, like when it is or isn't helpful to give a revision assignment. For example, tasks that are readily comprehensible aren't good candidates for revision.

Q: I just want to know what effective feedback looks like?

A: Here's what I'm proposing:
• Effective feedback doesn't look any particular way. From a picture, we can't tell whether we're looking at good or bad feedback.
• Revision of previously attempted work is a powerful tool for certain situations.
• Good feedback is whatever it takes to help kids improve their work, on their own, in class.
• We got into this mess, partly, by talking exclusively at a very high level of generality. Feedback is an enormous, fuzzy, abstract concept. We'd do better to bring our discussions down to Earth.
OK, and now I have some questions.

Q: What are some of the decisions that we face when designing a revision activity?
Q: Are there contexts in which different sorts of revision activities tend to be better?
Q: What sorts of situations tend to need whole-class feedback in order to help kids improve their work?
Q: What other concepts are there out there that are like "asking for revisions" in that they carve out a significant piece of the formative assessment landscape?
Q: What do students whose classwork was at a high level do during a revision activity?
Q: How do you support kids in a revision activity who need help but won't benefit from individual attention because of social reasons?

I have lots of questions, I can keep on going.

Q: What does effective feedback look like?

A: Effective feedback leads to revisions in class.

---
This is the seventh post in a series on feedback. To read the rest of the posts click here.

### When Not To Revise (Post 6 of 10)

Elixir or Water?

"Come, gather 'round! Everybody could use a drop of Gray's Fantastick Elixir! Drains the spleen, straightens your spine, furnishes beards and even improves digestion."

"Sir? Will your potion cure me of my ills?"

"Surely, son. The Fantastick Elixir is for all occasions. The more the better, the sooner the better!"

---

Nothing we do -- provided we do it with purpose and intent -- is always helpful. Medicine that heals everything really heals nothing. It's an elixir, a phony, or else it's a universal necessity of life, like water or air.

(Not that there's anything wrong with water, or air. But nobody needs to tell you to breathe.)

So too with learning. If some action always helps a student learn, in any situation, then it can hardly be said to help learning at all. It's a phony. Or it's something so basic to teaching or learning that it barely helps to mention it at all, like "listening."

I'm not sure that I can tell you when "giving effective feedback" does and does not help learning. I'm sure that it's because giving feedback always works, which means that it's a basic staple of teaching and learning. It's up there on the shelf with "listening well" and "explaining appropriately."

(Make sure you're drinking! Make sure you're giving feedback! Not everything that's crucial is important.)

On the other hand, I believe that I can say with some specificity when it does and doesn't help students to ask them to revise an assignment. If I can make this case -- and in this post I'll try -- then this should count in favor of "revising" as a concept that is important for teachers. Not because it's fundamental, but because it's not.

The Short, Simplified Answer

Stop here if you'd like. All that follows is evidence and argument.

Would Revision Help Here?

Rachel has been struggling with addition. She's a 4th Grader who -- until a week or two ago -- regularly got things like 6+4 wrong. The rest of the class is far more comfortable with addition/subtraction than she is, and one of the toughest challenges I face is making sure Rachel gets explicit practice with addition while still pushing the rest of my students.

(A recent compromise was to rejigger my units so that we began a unit on addition with larger numbers before diving back into multiplication. I reasoned that Rachel would have a better shot of hitting her trouble spots if I could focus on addition with the whole-class than if we flew on to a unit where addition wasn't at the forefront.)

When it came to adding two- or three-digit numbers, I noticed a few interesting facets of Rachel's work:
• When given a choice, she always prefers the standard algorithm
• When Rachel used the standard algorithm, she nearly always messed it up
• When Rachel was asked to use another strategy -- like breaking numbers apart by place value -- her work was very accurate

One day, while watching Rachel work, I noticed a mistake with her standard algorithm addition. No shock. So I asked her to use a different strategy, and there she goes, accurately breaking the numbers apart by place and adding. Huzzah. There's a discrepancy between her answers. I ask her which result she believes more. She points to her second, correct sum. Yay.

Rachel: "Yeah, but I really like stacking it."

(Teaching in a nutshell, right?)
So, what do I do? How can I help Rachel improve her adding?

A Cost-Benefit Analysis of Revision

Here's an option: I could pick a problem from Rachel's work, one she used the standard algorithm on. I could give her feedback -- written, oral, whatever makes sense. I can explain where she went wrong. Then, I could ask her to revise it, correct it and improve it.

Would that have been the right call?

I don't think so. Much better, I think, to give Rachel feedback and then a new problem to work on.

Why? There are risks -- in the scheme of things, relatively minor risks -- associated with asking Rachel to continue to work on a problem that she already completed.
• Engagement Risk - If I ask Rachel to continue working on a question that she already answered -- even one she answered incorrectly -- I run the risk of boring her. After all, she's already seen it before, and it's new things that tend to excite our students. And maybe she won't give the feedback credence at first because she thinks that she already got it right...
• Social Risk - ...or she'll see that it's wrong and resist engaging because she doesn't really want to feel dumb. Kids like improving, but there's always a risk that feedback will go wrong, or that the personal attention of a one-on-one meeting will be embarrassing.
• Learning Risk - Another way that this whole thing could go wrong is if Rachel makes the (relatively simple) local correction without actually improving her addition skills. There's not exactly a whole lot to correct or revise in a single addition problem. She's going to need a few examples to improve.
In many situations, these risks are worth taking. Why? Because it would be incredibly costly to assign a new problem to students. We don't want students to spend class time making sense of an entirely new context just so they can tweak one small (but crucial!) detail. And making new problems that target specific areas is time-consuming for teachers. And maybe something about the new task will be more complex than intended, and the conversation won't focus on the area of need.

When revision works, it works because we can get to the point. Yes, we know the context. We understand what the problem's asking, and we even know a lot about how to find the answer. There's just this one area we can improve. This one thing. Let's make this better, I'll help you.

But addition problems? Shoot, I can make 'em up on the spot, and Rachel won't need help understanding what I'm asking her to do. There's little to gain from asking Rachel to revise these sorts of problems.

Revision Is A Sometimes Thing

So revision -- even done well -- isn't an always thing. It's not a staple. It's not a panacea or an elixir either. (And that has got to be at least part of the reason why we don't talk very much about it.)

Revision is a thing that sometimes helps. We constantly have to weigh the benefit (skipping to the point) with the risks (engagement, social, and learning). It won't always help, and that's a very good thing indeed.

Appendix: What I Ended Up Doing For Rachel

I wrote up this task. I told the class that there was a kid named Joe and that Joe's method for adding was to just add the digits in each place together and then write them down one after the other. (Rachel: "I use Joe's Method a lot.")

You shouldn't take my word for it, but it went well and her work has improved.

---
This is the sixth post in a series on feedback. To read the rest of the posts click here.

## Saturday, November 22, 2014

### Sooner is Not Better, and There Is No "Best Feedback" (Post 5 of 10)

At the start of this series, I highlighted four claims about feedback that I thought were both widely heralded and wrong.
1. Oral feedback is always better than written feedback.
2. When it comes to feedback, the more the better!
3. When it comes to feedback, the sooner the better!
4. Feedback is all about helping students understand the mistakes they've made.
Since then, I've shared a case when I gave oral feedback and a case when I gave written feedback. In one case I gave lots of feedback and in the other I gave rather little feedback.

Sometimes oral feedback is better than written feedback. Sometimes it's the other way around.

The point isn't just that these myths are wrong, though they are. It's that this talk of "best practices" (or "best activities") is such a limited and unhelpful way of talking about teaching.

In this post I'll argue that sooner isn't always better. Then, it's time to put these sorts of claims to rest. There's a better way to talk about teaching.

Point: When It Comes To Feedback, The Sooner The Better

"In most cases, the sooner I get feedback, the better." (Seven Keys To Effective Feedback)

"Of course, it's not always possible to provide students with feedback right on the spot, but sooner is definitely better than later." (5 Research-Based Tips For Providing Students With Meaningful Feedback)

Counter-Point: Nah

A few weeks ago I gave a quiz to my geometry classes. We were at the end of a unit on quadrilateral properties and proofs, which was the area that my quiz targeted.

When looking at the quiz, I noticed that my three "always/sometimes/never" questions had gotten a huge variety of responses: "always, always, sometimes"; "sometimes, sometimes, it depends"; "sometimes, always, never", etc.

A closer look confirmed that their thinking was all over the place. A handful of kids indicated that parallelograms have a line of symmetry. Others didn't recall that a kite can be split into two congruent triangles. Others thought that a trapezoid's diagonal divides it into two congruent triangles. Some proofs were nice, others needed lots of improvement.

I quickly decided to give my class feedback and time to revise: no other activity that I could run in class would be able to address each individual mistake and give each student a chance to think about their specific area of need.

I decided to wait a week.

Why wait? It wasn't because I didn't have time. I think that waiting was the right decision. Consider the work of one of my students, Jake:

Jake showed some major limitations in his thinking on this quiz. Jake indicated that only parallelograms can be split along their diagonals to form two congruent triangles and that the consecutive angles of a parallelogram don't sum to 180 degrees.

Jake was hard-working during class, and he participated actively in many of our conversations in class about parallelograms. In short, he hadn't slacked during the past two weeks. Despite this, he was still having trouble putting the pieces together. Would feedback help him where two weeks of instruction couldn't?

Maybe. But here's another thing I knew: the upcoming week quite possibly would help. Why? The unit we were beginning was studying how one shape (e.g. a parallelogram) can be dissected and rearranged to form a new shape (e.g. a rectangle). The activities that I was planning on running would give us a chance to physically rearrange shapes into other shapes. I thought that these activities would probably give me a number of great opportunities to address some of Jake's misconceptions about congruent triangles in kites and parallelograms. Or, at least, we could draw on these dissection examples when discussing his quiz when I did give Jake feedback.

Another factor: Jake was hard-working, but I saw a drop in engagement after a few days of "always, sometimes, never"-style questions. I wasn't shocked. We had been working on them for a few days, and questions that are posed in a similar fashion can start to bleed together after some time. I thought that Jake (and others) needed a break from these sorts of questions.

So I waited a week, and I gave feedback. And it was fine.

OK, Great. Every Rule Has Exceptions. Including "The Sooner the Better." Who Cares?

Thanks for the question, sub-heading!

It's true: there are certainly times when the decision to wait a week to give feedback is the wrong one. And is that the majority of situations? The vast majority?

I don't know. I do know that there are lots of situations like my quadrilaterals quiz, and there are a lot of students like Jake. And that "the sooner the better!" is, strictly speaking, false.

Here's the only question worth asking, then: do we learn much from a slogan like "the sooner the better"?

I don't think so.

We don't need "5 Research-Proven Teaching Techniques" or "7 Activities For Learning" or "12 Qualities of Excellent Feedback." Ultimately, the work of teaching comes down to the decisions that we make, and writing about teaching should be about improving those decisions.

What we do need is to talk about the dilemmas that we teachers commonly face, and try to find guidelines that help us make wise decisions in those scenarios.

Talk of "the sooner the better" isn't anywhere close to that. In the rest of this series, I'll try to talk more productively about decisions we face when giving feedback for revision. The stories that I've shared so far are my attempt at a start to that work.

---
This is the fifth post in a series on feedback. To read the rest of the posts click here.

## Saturday, November 15, 2014

### Against the Feedback Menu (Post 4 of 10)

So you're giving a presentation to teachers. It's called "Giving Effective Feedback to Students." You've got slides. You've got an audience. You've got 60 minutes.

How do you spend it?

The way we teachers talk about feedback, you're likely to present about the menu of options that teachers have for giving feedback. You'll talk about written vs. oral feedback; positive vs. negative feedback; individual vs. whole-class; timely vs. delayed feedback; lengthy vs. brief feedback; feedback for learning vs. feedback for evaluation.

That's certainly what's going on in Types of Feedback and Their Purposes:

Here's an opinion about how we talk about feedback: this is an insane way to talk about feedback.

What makes this crazy is the extraordinarily high level of abstraction. It's so high-level that it's practically philosophical: "What is good?", "How should we write?", "What's best to do?"

In a different planet, we would talk about teaching situations and how to improve them. That seems more sensible to me than talking about things that improve learning and then matching them to various scenarios, post-hoc.

By analogy, imagine that I was presenting on "Giving Effective Drugs to Patients" instead of "Giving Effective Feedback to Students."

My first slide would, of course, cite a relevant research table showing -- conclusively! -- that giving drugs helps patient outcomes.

Of course, first we would have to define "drugs." (The first sign that we're dealing at too high a level of abstraction -- we don't even know what we're talking about!)

Too many doctors just prescribe Asprin for everything, don't you find? (Compare to: "Feedback vs. Advice".

Now, time to dig into the details! Doctors have to make important decisions about the amount of drugs, as well as the timing.

And don't forget - no two patients are the same!

My point here is not to compare teaching to medicine. They're very different fields! I'm making a more limited analogy between the way we talk about instructional techniques and the way doctors talk about medical strategies.

"What is the most effective way to use drugs?" is an incredibly general question, and not a particularly helpful one. A better question to ask would start with the ailments: "What's the most effective way to treat whooping cough?" or whatever.

So too in math education. The "menu of options" style of presentation is pervasive in talk about teaching, but it doesn't have to be. A more productive route, I think, would be to start thinking about various specific teaching-scenarios that are common throughout the profession. How do we make good teaching decisions in these scenarios?

What sorts of scenarios am I talking about? Here are two that I'm spinning off of earlier posts in this series.
• Toni's Fixed Mindset: How do you help a talented student who is socially unable to admit to any mathematical limitations to learn a complex skill, like proof? (Post #3)
• Misinterpreting Bar Graphs: How do you best help a class that has a wide variety of minor, but significant misconceptions that can't be addressed all it once through whole-class instruction? (Post #2)
In the case of Toni, I argued that the best feedback was whole-class, oral instruction that lead to students choosing just one solution to revise. In the case of the bar graphs, I argued for giving individual "highlight"-style feedback and group-time to revise the entire assignment.

Written vs. Oral? Whole-class vs. Individual? Immediate vs. Delayed?

How much good can come of these questions?

---

This is the fourth post in a series on feedback. To read the rest of the posts click here.

## Tuesday, November 11, 2014

### Letter to A "Second Drafts" Skeptic (3 of 10)

Dear Michael,

I read your previous post and was intrigued by your argument. I'm sympathetic to the idea that "giving feedback" is hopelessly vague. And, I agree: giving feedback is only sometimes effective.

You suggested that teachers need to take the energy they put into "giving feedback" and redirect it into "facilitating second drafts."

I'm skeptical.

I'll grant that your post showed an example of "second drafting" that led to student learning. Sure. But who says that "improving the classwork" is ever the most effective activity to get to this learning? After all, the effectiveness of a teaching technique is at least partly determined by how much time it takes. Also: maybe you got lucky? Maybe asking students to improve their work wasn't the right decision to make, even if it happened to lead to learning.

In short, you've got more work to do.

Sincerely,

Skeptic

---

Dear Skeptic,

I agree completely. Here’s what I’ll try do to address these concerns. First, I’ll argue that there are at least some situations when helping students improve their classwork is the right instructional decision. But it’s a long way from saying that “second drafting” sometimes works to saying that it works in most classrooms, in many situations. I won’t be able to bridge that gap entirely, but I'll try to get as close as I can.

To start, I want to tell you about an awesome student of mine, Toni.

Toni is a 9th Grader. She's new to our school, and she stood out early on. On the first day of class, I gave students a version of the handshake problem and put them in groups. Toni took a quick look at the problem and said, "Oh this is simple" to her groupmates. She took a marker and whiteboard and scribbled a few calcuations without a word of explanation to the folks she was working with. She quickly and accurately calculated how many handshakes we’d need to all shake hands (but didn’t try the second half of the problem in search of a generalization) and sat back and called me over.

"We're finished."

It was only after I nudged them that Toni (and her group) really dug into the second problem.

This is the sort of student Toni is. Quick, accurate, sharp, and likes being quick, accurate and sharp.

Toni has a group of friends in class that she gets along well with. One day I overheard them talking about how smart they were at math. Toni said, "Oh yeah I'm really good at math. It's in my blood." Being good at math is clearly an important part of Toni's identity. She sees smartness in math as something pre-determined and relatively fixed.

Another thing about Toni: she often didn’t justify her work, even when the assignment explicitly called for justification. In her classwork or quizzes, Toni would quickly come to conclusions that were poorly justified. Often (usually) they were dead-on. But sometimes they weren’t. For instance, for several days Toni was absolutely sure that she had found a counter-example to SAS.

I tried to give Toni some one-on-one feedback on her SAS counter-example during classwork. It was hard going. She didn’t seem interested in what I had to say, was easily distracted. I tried to get her to use reasoning to check her conclusion, but her confidence seemed absolute.

This pattern continued over the course of a few weeks. Toni would produce work quickly and without proof, even though we talked about and shared proofs explicitly during our class discussions.

This came to a head on a classwork assignment about trapezoid properties. My major goal for the lesson was to help students understand how to use parallel lines and triangles to write proofs about quadrilaterals. I gave students a problem set including the question, “Is it always, sometimes or never true that the sum of the angles of a trapezoid is 360 degrees?”

I came over to Toni and saw that she had written “always” by it. I nudged her to offer an argument, and she wrote “Because all quadrilaterals sum up to 360 degrees.” I asked her how we knew that, and she wrote “y=180(n-2)” on her page. Eventually, with a bit more nudging, she wrote "alternate interior angles are supplementary."

During my initial instruction I had suggested that students try to use parallel lines and alternate interior angles and the decomposition into triangles to develop these trapezoid proofs. The day before, we had worked on a proof that triangle angles sum to 180, and a student had shared a proof using parallel lines. I tried to use questioning to push Toni in this direction, but even as were talking she was looking at other problems and patiently waiting for me to leave her alone.

In other words, my initial instruction failed.

Now, Skeptic, how would you have me react to this failure of initial instruction?
• Explicit instruction in a whole-class setting had already failed, and it’s not hard to understand why. Toni was heavily invested in being especially good at math, and was heavily incentivized to avoid making a connection between corrections in instruction and her own work.
• I could create a new task whose sole purpose was to focus on the need to justify our claims in geometry. This would suffer from the same issues as explicit instruction, though. A further issue: the new task would, by definition, have new intellectual demands that would distract from the old ideas that I wanted Toni to learn. Any new math would be something else for her to think about rather than laser-focusing her on her areas of improvement.
• I could try conferencing one-on-one with her during class, but this had already shown itself to be problematic. Her status/mindset issues suffice to explain why this had failed. Toni has trouble yielding when in the presence of her friends and peers.
• I couldn't meet with Toni outside of class. She didn't see herself as needing my help, and the logistics of arranging an out-of-class meeting are difficult for us.
• What about simply giving feedback without time to improve her work? This feedback is easily forgotten and ignored, in general, and wouldn't have been taken to heart by Toni, who has shown herself to be especially unreflective when it comes to proof and argumentation.
I decided that the only course of action that had a chance of working was giving feedback to the whole-class and asking the entire class to improve their earlier work. This way, Toni wouldn’t be able to escape from the connection between my instruction and her own work. She wouldn’t have any other mathematical distractions -- any other problems for her to show her ability on by rushing on to them during instruction. Most importantly, since the entire class would be improving their work, she wouldn't feel as if her struggles were on display for others. Toni could have a chance to quietly make improvements without losing status in the eyes of friends.

Here's how I did it: I handed everyone’s papers back with no comments, to minimize everyone's ego-involvement. Then I posted on the board: “Is it always, sometimes, or never true that a kite can be divided into two congruent triangles?”

After discussing the problem and sharing proofs verbally, I told the class that I had read and enjoyed their work from yesterday, but had noticed that lots of them had room to improve their proofs. I said that proof-writing is new to a lot of us, and is a different skill than answer-getting in geometry. We can get better at both, I argued. (At this, Toni groaned.)

Then I shared three “sample” responses to the kite problem:

Not a proof. No reasons given

OK proof. Reasons given, but they're private reasons

Great proof. Reasons given that others can understand.

After giving this feedback, I said: "Pick one of your problems from yesterday, and write a nice, improved proof, like this third proof, on a new piece of paper."

And then...

---

OK hold on a sec. This is Skeptic again, and I know where this is going. "And after this, the look of understanding! a smile blooms on Toni's face! She laughs and says, 'Mr. Pershan, I didn't realize that I can do this!'"

Right. And from that day on, Toni's proofs were much improved, and her attitude in class was better, and then five years later she calls to tell you how important your class was to her.

Pardon me, but I'm skeptical. I don't know Toni. No one knows Toni. You know Toni, and for all we know you're making this up.

You expect us to believe your "second drafts" idea based on your totally unverifiable story?

Signed,

Skeptic

---

I don't expect anyone to believe me based on my word. You shouldn't. I'm probably not making this up, but I'm also not an objective observer of my own classroom. Further, I can't be sure that what I think caused Toni to improve actually caused her improvement.

Instead, inspect my thinking. I'm not claiming that asking Toni to improve her work caused her to learn how to write a proof. I'm claiming that this was the only sensible teaching decision to make in this situation. Is there a flaw in my instructional reasoning? Is there an alternative to classwork-improvement that I didn't consider? Do you think that one of the solutions I dismissed could have helped?

If my reasoning is sound, then the first half of my response to a skeptic (and Skeptic) goes like this: this is a type of situation where asking students to improve their work is the best teaching decision. When a student's struggles aren't even on his own radar, asking for a second-draft is often going to be the best course of action, in particular when that student is disposed not to see their own areas of improvement.

But the skeptic's gap remains: how do we know that asking for a second draft is a generally useful teaching move, and not just helpful in the specific scenario that I outlined above?

The best I can personally do is share my hunches and speculations. But that's another post for another day.

Questions

I love your comments, and I have a special love for your skeptical comments. Here are some questions I'm thinking about after writing this post.
1. Do you think that asking students to improve their work is generally helpful for learning? Why do you think so?
2. In some ways, this was pretty cautiously argued. Is this sort of justification of teaching decisions necessary?
3. Should we believe the evidence that teachers report from their classroom? Would you have been convinced if I had just said "...and now Toni is writing amaaazzzing proofs"?
4. Is there still room for the skeptic?
Appendix

As it happens, Toni's proofs have been much better.

And for those playing at home, cross off "whole-class," "verbal" and "goal-setting" on your feedback Bingo card.

This is the third post in a series on feedback. To read the rest of the posts click here.

## Saturday, November 8, 2014

### Kids Learn From Second Drafts (2 of 10)

In this post, I'd like to argue that "giving feedback" is a lousy teaching concept. I'll suggest that instead of worrying about how to give effective feedback, we'd be better off thinking about how to help students improve their work in a second draft.

I've pointed out that there is significant, widespread disagreement within the teaching profession about the usefulness of feedback to learning. And I think that "feedback" only has itself to blame. It's a famously difficult concept to define, probably due to the convoluted route it took from its origins in electrical engineering through group dynamics, finally landing into the psychology of learning and then into popular usage. Along the way, "feedback" went from meaning something precise ("when the output of a system in turn becomes an input") to something vague ("when a person says a thing to another person about some thing that the first person did").

Because of the vagueness of the term, to tell a teacher "give some feedback" gives him practically no guidance. I think that more helpful advice would be to tell a teacher, "help the kids to improve their work," or "have them do a second draft of their classwork."

Consider a quick scenario. Suppose that you're teaching eleven 3rd Graders how to read bar graphs. You finish a first round of instruction and give them this bar graph and some questions to answer.

You collect their work, and read their responses in more detail. Their thinking is varied. Some ideas are mistakes, and (as always) there are aspects of their work that could be better. In particular...
• Some students looked at the gap between 180 and 200 and concluded that there were only 20 non-walkers from Parks School.
• In response to "How many students either bike or walk to Lincoln School?" some students wrote "60 and 40" instead of "100."
• When prompted to explain some of the differences between these two schools, some kids stop short of conjecturing what the underlying differences between Parks and Lincoln might be.
A teacher who wants to help these kids by "giving feedback" might do any number of things. She might go through the questions one by one with students in whole-group. Or he might indicate right/wrong on the page and ask students to make corrections. Or she might make the corrections on the page for the student. Or he might ask students to check each others' work.

You might start wondering, what are we trying to accomplish with this feedback? When does it lead to learning? When is it a waste of time?

Here's my motto: Kids learn from improving their work, and effective feedback is whatever it takes to make that improvement.

What I ended up doing with this 3rd Grade class was highlighting in blue something that I thought was really great in their work, and something in yellow that I thought they could improve. I handed back their work and let them work with partners on improving (and finishing) their assignment while I circulated and had conversations where I checked in.

In the course of things, a "20 non-walkers" student understood had a chance to think things through and improve their answer.

Students who didn't know how to explain the data had a chance to improve their work.

The feedback that we'll end up giving is going to vary drastically depending on the math, the students, and the teacher. I give written feedback, verbal feedback, whole-class feedback, individual feedback, small-group feedback, questions as feedback, observations as feedback, advice as feedback.

The only non-negotiable here is the goal. The goal is to help kids see ways that they can improve their work and then to give them a chance to do so.

At this point, a good skeptic might be doubting whether a student improving their work really leads to learning. Or whether creating a second draft is more effective than other forms of instructional activities, like directly telling or working on a task that's related to the first. These are good, important questions.

In the next post in this series, I'll make the case that there are times when improving student work is the best instructional move available.

---
This is the second post in a series on feedback. To read the rest of the posts click here.

## Tuesday, November 4, 2014

### The Four Myths of Feedback (1 of 10)

Is this good feedback? Circling the areas of mistakes, highlighting areas of concern, debugging students' work?

Is this? Asking questions, explaining the mistake that a student made?

More puzzles:
• Why do some teachers report that their students just crumple up their feedback, while others teachers swear by the feedback cycle?
• Why do some teachers report giving written feedback to be inefficient and exhausting, while others find it manageable?
• How can we bridge these huge gaps in teacher perceptions? Why hasn't research settled these questions for teachers already?
My hypothesis is that we're talking past each other. Do kids like feedback? Which kids? Is written feedback exhausting? Well, what sort of feedback are you talking about?

So much of teaching is in the details. Unless we give these details in our writing about teaching we can hardly be sure what we're talking about. Without classroom evidence, discussions get very fuzzy, very quickly.

In this series of posts I'm going to use stories from my classes to argue for the viability of a certain kind of feedback. In particular, I have my aim set against four claims about feedback that I consider to be myths.

Without further ado, here they are, your myths!

Four Myths of Feedback
1. Oral feedback is always better than written feedback.
2. When it comes to feedback, the more the better!
3. When it comes to feedback, the sooner the better!
4. Feedback is all about helping students understand the mistakes they've made.
Don't agree? I'm not surprised! These statements have been presented entirely without context or grounding in any classroom. My job over the next 9 posts in this series is to put flesh on the bone of these claims with some believable evidence from my classroom.

You have two jobs. First, to be productively skeptical. I'm going to be providing you with enough detail that you'll be able to disagree with my interpretations of the classroom. As far as I can tell, this is rare for a writer about teaching to do, and I'm looking forward to our disagreements.

Your second job is to think of stories from your own teaching that might add to the feedback picture. My classroom has its own particular constraints. So does your's. We'll learn a bunch from the collage that emerges from the volley of classroom stories.

This is the first post in a series on feedback. Once they're all written I'll link 'em up here at the bottom of the post. In the fuuuutre.

Update: Welcome to the future! Here's the whole series. (link)

## Sunday, November 2, 2014

### Focusing On Feedback

I want to try a bit of a writing experiment. The next ten posts that I write will be about giving feedback. The plan is to stray away from abstractions and research summaries. Instead, I want to make the case for the decisions that I'm making in class through stories and analysis.

So, stay tuned!

## Thursday, October 23, 2014

### A Time When I Directly Told

We’re studying data in my third grade class. My students as a whole came in with vague notions of the meaning of data (“it’s information”) and some kids were confused as to what this all has to do with math anyway. On an initial dataset, kids mostly categorized things by their superficial features (“restaurants go together”) instead of grouping data more purposely in order to answer particular questions. Their descriptive language was mostly limited to “most” and “least”, and the questions they posed reflected that.

OK, that’s the preamble.

The main task today related to a survey that we took of our class. The survey was titled “Places where we like to...”, and we collected information about where everybody in the class likes to do various things. So: “Places where we like to read.” “Places where we like to visit.” “Places where we like to eat.” etc.*

* (The full lesson plan is here, at the bottom of the piece.)

Anyway, a pair of kids were organizing the class’ data on “Places where we like to read.” This was the data set they were organizing:
• School
• In my room, in bed
• Public library
• Home, in bed
• Library
• In the living room in my house
• My bed
• bed
• Home
• Home
• Home
• In my bed with the puppy
When I came over, they had organized all the data points into two categories. They had the data on little cards, and these were their groupings:

Group 1:
• School
• Public library
• library
Group 2:
• In my room, in bed
• Home, in bed
• In the living room in my house
• My bed
• bed
• Home
• Home
• Home
• In my bed with the puppy
I asked the pair, “What are your categories?”

“This one [on the left] are places for education.”

I was excited by this, because up until this point it had been very difficult for the students to see any other possible organizations of the data that didn’t just categorize their surface features. Actually, this was their second attempt at organizing the dataset, and their first had been fairly typical: all the “home” was grouped together, all the “bed”s were grouped together, “school” was alone, “library”s were together, etc.

In other words, I thought that I was watching a moment of learning, where the kids saw what we mean by organizing data to answer different questions.

So, I continued probing: “Got it. And what’s the other category?” I pointed to the right grouping.

Student A: “This cateogry is in my room, in bed, and also in the living room...”

Student B: “In home, a bunch are in home...”
The kids continued to trip over each other in an attempt to fully characterize their second category. I saw this as evidence that they recognized the insufficiency of their categorization, but lacked the language to properly describe it.

This, I recognized, was a perfect opportunity to tell them something.

I said: “Oh, so this category on the left are the places where education happens. And this, on the right, these are not places of education.”

Student A: “Yeah like a home or ...”

I continued: “So you might say that this category is places of education, and this category is places of non-education. Did I get your categorization right?”

The kids nodded. One of them repeated the categories using this new language, and then later used it when I brought the class over to look at the categorization that this pair had made.

I think this was a time when directly telling was able to cause learning. I’d generalize this by saying that when kids are searching for a term and can’t find it, that’s an opportunity for causing learning by directly telling. I’d also argue that you can cause learning by engineering these sorts of moments.

Questions:
1. Do you think the language of "direct telling" is appropriate for
2. Are you convinced that this was an appropriate time for direct telling?
3. What parts of this post helped you feel like I wasn't bamboozling you in my representation of what happened in class?
4. Were there moments while reading when you found yourself wishing you could have seen this moment? Where in the post were those moments?

## Wednesday, October 22, 2014

### How To Not Quit Teaching

Thinking of leaving teaching? Here are some posts to get you started:
Read them all, and you'll find yourself with many arguments for leaving classroom teaching. But what if you want to settle your doubts and stay? What's helpful for getting past this sort of angst, besides leaving?

These days, I'm feeling pretty good about classroom teaching as a long-term gig. Personally, I found it helpful to interrogate some of the assumptions behind my earlier angst about teaching. Here are a few disorganized, stray thoughts that came out of my reflection:
• All my angst was premised on the attractiveness of classroom teaching. I wanted to teach.
• I realized that, right now, I have enough money. If I was careful, I could probably continue to have enough money. My angst wasn't premised on concern for material well-being.
• Mansions are cool, but drafty in the winter and I can hardly keep track of my keys as is.
• The amount that a profession gets paid is confusing. What does it mean to say that a certain job is overpaid or underpaid? Do I deserve to make more than a nurse? Than a secretary? As much as a doctor?
• Teaching is a profession that is currently and historically dominated by women. A lot of the teacher-administrator relationship is premised on the idea that women should be accountable to men.
• Eventually, I concluded that the way I felt about teaching was the way women often felt at work.
• Fundamentally, I like learning and challenges. No one is stopping me from figuring out how to keep learning about teaching. I can direct my own learning and set my own challenges.
• The problems that I have in teaching, then, are the problems that I would have in any profession. If I were a doctor, I'd need to figure out how to keep learning in the face of comfort. If I were an educational researcher, I would need to figure out how to direct my own learning.
• In that case, I had better figure out how to direct my own learning.
I propose that, from the standpoint of current classroom teachers, it's more productive to think of how to guide one's own learning than to worry about the state of the profession.*

This isn't to say that it's illegitimate to worry about the state of the profession. Activism is cool. So is being informed. All I'm saying is that "OMG THE SKY IS FALLING AND TEACHING HAS NEVER BEEN WORSE" isn't a particularly helpful perspective for me to dwell on, because I have/want to be a classroom teacher.

Update: More "I quit!" posts.

## Sunday, October 19, 2014

### An Open Comment Thread

Mostly for me, but feel free to jump in.

## Sunday, October 12, 2014

### Thinking Through a Decision: Tom and Addition

The Class, The Kid, The Activity:

I'm going to share with you a situation from my 3rd Grade class and a decision that I made. Then, I'm going to ask you what you'd do in this situation.

This post comes with a question and a meta-question. The question is, "What's the right teaching decision?" The meta-question is, "How much context needs to be shared so that we can ask useful teaching questions?"

I'll attempt to provide context at three levels: about the class that I'm teaching, about the kid that I want to focus on, and about his work on a particular activity.*

* I'm unsure how important school context is to this. Briefly: I currently teach at Saint Ann's School, a private school with competitive admissions in New York City.

The Class: 3rd Grade Math

My 3rd Grade class has 11 students. My school has about eighty 3rd Grade students, and about half of them are in a "fast-paced" track and half of them are in a "regular-paced" track. This class is a regular-paced class. We meet less often that most math classes meet: 4 times a week, 35 minutes a day.

We've been studying addition for the first five weeks of school as part of our first major unit. (I have lots and lots of curricular freedom, but I've elected to mostly use the TERC Investigations activities so far.)

Here's a sample of some of the work that we've been doing in class over the past few weeks.

In addition to our main activities, we've often been starting our sessions with number talks aimed at helping students build a greater facility with efficient mental addition.

The Kid: Tom

Tom is a 3rd Grader. (Tom is not his real name, but privacy etc.) Tom has worked slower than other students in class on most problems that we've done this year. I randomly call on students fairly often, and when Tom's name gets picked he often struggles to articulate a line of reasoning.

On independent work, Tom works slowly but carefully. He often articulates his reasoning clearly using equations. Here's a sample of Tom's independent work.

Tom has had moments of great thoughtfulness over the first few weeks of school, and it's been fun to watch him make progress. In sharing his reasoning he sometimes makes mistakes ("65 cents and 25 cents makes a dollar") but when given a chance to slowly think it through he's able to improve his thinking ("No that can't be right because 65 and 20 make 85 and 85 and 5 make 10...")

Another nice moment from Tom: he solved 27 + 8 mentally by breaking 8 into 3 + 5 and "making a 10" with 27 and 3.

In short: Tom uses a variety of strategies quite well, but his addition is inefficient and error-prone.

The Activity: Formative Assessment for Addition

After a few days of some rich and interesting number talks, I wanted to know what addition facts my students still struggled with so that I could properly steer things in the next week. I knew that a few of my kids, including Tom, still struggled with mental addition, but I didn't have a good understanding of the types of problems that they struggled with. To assess which addition facts my kids "just knew" and which they had to figure out, and also whether they could figure them out, I decided to give my class 33 addition questions and ask them to work on them alone.

Knowing that these sorts of activities sometimes cause anxiety in students, I was careful in how I framed the activity for kids. I didn't tell them that the activity was timed. I took care to make sure that there was another activity ready for students who chose to move quickly that they could silently access. I also made it clear that I expected there to be questions that they didn't "just know."

Here's a video of how I introduced the activity.

Here's Tom's work on this activity.

(About the stars: instructions to the class were to "start with the ones that you 'just know' and put a star next to any that you need to think more carefully about, and then work on those. By the time I noticed that Tom wasn't returning to the ones he skipped, I decided that this would be helpful information anyway and there was no need to have him work on these problems. That might have been a mistake.)

I noticed that Tom has very little problem with adding 9's. Knowing his sometimes-facility with certain addition strategies (and his lack of quick recall of addition facts in general) I think that Tom is reasoning relationally here, e.g. 9 + 7 = 10 + 7 - 1. This is nice.

I noticed that Tom skipped lots of questions that involve adding 8 or 7. I noticed that he didn't "just know" 3 + 7, which might have been a useful fact for figuring out 7 + 6 or a lot of the other 7's.

Tom's work wasn't typical of the class'. Most students slowly, but accurately, answered all the questions. Casie's work (below) was more typical of the class. (The rest of the class' work is here.)

The Question and a Decision:

I've had students like Tom in every class I've ever taught. How do I help Tom while still pushing the rest of the class forward?

After thinking things through, here's what I decided:
• It seems to me that Tom could use some more strategies. In particular, there's a handful of additions that he could know very quickly if he started seeing 6+7 and 8+7 as "near doubles."
• Tom could also answer a lot of the problems that he missed using a "make 10" strategy. But in whole-class situations he's already used this strategy. Why didn't he use it on this assessment? He might just not have made the connection, or he might not "just know" the numbers that combine to make 10. (He missed 3+7 on the assessment.) I also think that "make 10" requires you to hold a bunch of numbers in your head, and that this strategy tends to be more cognitively demanding than other strategies. He might not be ready for it yet.
• Tom needs a chance to rethink the questions that he didn't get and make it an explicit goal to improve on them. Here's how I'll do it: I'll do a number talk that makes some of the useful strategies explicit, then I'll return their assessment work and ask them to work on any of the questions that they skipped or found difficult. I'll suggest that they use some of the strategies that were just discussed. (An issue is that many kids won't have any problems to work on, and I don't want to draw a distinction between those who do and those who don't. I'll mitigate this by giving everyone their work to improve and a follow up assignment that they'll work on individually and silently.)
• Over the next few weeks I need to give Tom an opportunity to practice and learn some of these additions that he struggled with. I'll do this mostly in the context of number talks, because these are easily differentiable. I can put four problems on the board that increase in difficulty, and ask students to mentally work on all four. This could give Tom a chance to practice "6+8" while giving Casie a chance to try "46+68" in her head.
I'll want to check back in a few weeks to see what sort of progress Tom (and others) have made.

In Conclusion:
1. What here would you do differently?
2. Should I be giving more individual feedback to Tom? How would you give it?
3. Why else might Tom be having trouble here, besides for the ideas that I suggested?
4. Missing from all this is the social aspect -- how Tom interacts with other students in class. How do you think I could represent this?
5. Oftentimes I'm unconvinced by a piece of writing about teaching because I worry that it's an unfaithful representation of the classroom. Do you feel convinced that this representation of my class is faithful?