Transformation Feedback that Worked, I Think

I felt good about how I handled feedback today in class. The class has been studying transformations in 8th Grade, and I wanted kids to practice visualizing different transformations. The task was to match shapes with transformations.


The above work is something I saw a lot. My kids had a pretty good sense of the different motions that translations, rotations and reflections were describing, and overall were pretty accurate.

Rotations around centers besides (0,0), though? Kids didn’t seem to know how to handle that. Some treated a rotation around (2,0) as just another rotation around the origin. Others seemed to get the quadrant of the image correct, but the coordinates weren’t right.

I decided that this would be what I’d focus the start of my next lesson on. I’d start with a pre-feedback activity, give some quick comments on the work, and then ask kids to revise their work.

This was the diagram I started class with:


I was trying to figure out what prompt to start with. I thought about just asking kids, “Where would the image be?” but this idea had problems. First, I wanted to know a bit more about how they thought because I wasn’t sure exactly where the problem was with their thinking about this. Second, I knew that kids would have trouble with this based on their previous work.

Instead, I posted the image and asked kids to figure out as much as they could about the diagram. I figured that this vaguer prompt would be more revealing while also giving me a chance to respond to ideas and help kids learn how to think about this sort of rotation.

(I also thought about just telling kids, here’s how you do this. I think that would have been plenty effective in this case, but I also like giving kids chances to have interesting mathematical ideas, and this seemed like a chance for that.)

Here was the annotated picture from under the document camera.


The three ideas on the top were from kids. After hearing them I pointed out that some of these were correct, assuming that the transformation was around the origin.

Then I said, this is why I brought this image into class, because I noticed from your classwork that rotations not around the origin are hard. I have something that I think will help, I call it drawing a tether. And then I connected points A and B. Does this help you figure anything else out?

Very quickly a kid said, yeah, now I’ve got this and there was a lot of chatter about where the other points would go. I annotated those, and summarized that adding the tether would help. I also said that you might figure out where the other points are relative to B, and this led to more mathematical chatter and more asserting of where the points are. (Is that the sort of evidence I can use to know if a class is working out? Mathematical chatter?)

Then I said, I want you to try out these ideas while revising your classwork. I highlighted ones that could be revised, and I’d like you to draw a revision on the page. From past experience, I wanted to keep the timing of this quick, so I said you’ve got 5-10 minutes, and if you don’t start quickly then you probably won’t get a chance to learn much now.

Here was some of the revised work.


Not everyone got feedback about rotation around (2,0). These kids get feedback about reflections around y = x and y = -x (which is good, because we need to know those for the next activity we’re working on).


The questions that I’m left with are about how sure I can be that this lesson went as well as I felt it could be. What’s my evidence? Is it the way the kids were acting? Was it their engagement? Or their resulting work? Should I have done a quick exit assessment — would that have supported my claim that this worked?

And does “this lesson worked” mean that it just worked in the classroom? Or can it mean that the planning worked, that I made moves on purpose and with confidence?

How do you really know if a lesson worked?

A Difference Between a Number Talk and a Number String

Suppose you’re working on adding multiples of 10 with your 3rd Graders. (Suppose, further, that you should be planning that lesson instead of writing a blog post. This will only take a second though, I swear!)

Here’s the thought. A lot of kids have trouble with 67 + 40 but can handle it without trouble if you encourage them to add a bunch of 10s.

Our ability to add 67 + 40, before it’s second-nature, can come from being able to see this essentially static expression as rolling out in time. In other words, we can see “add 40” as “first add 10, then add another 10, do it again, and finally add 10.”

And I think a lot of learning is like this. There’s a process or procedure that we can do, perhaps, but it takes a lot of time. So we get good at that procedure, and then we summarize it in our minds. We can then see the summary as representing a process that rolls out in time. And eventually, the hope is that we don’t even need to remind ourselves of that process, and the summary suffices.

ANYWAY, a number string is an opportunity to work on the process, and even to summarize that process. For example:

67 + 10

67 + 10 + 10

67 + 20

67 + 10 + 10 + 10

67 + 30

67 + 40

But once kids are able to handle this comic-strip form of thinking about 67 + 40, we would want to give them a chance to mentally roll out this procedure when prompted with just a summary of it. So, for example:

67 + 10 + 10 + 10 + 10

And eventually we don’t want to offer anything other than the problem itself in its least dynamic, it’s most out-of-time form. Which is 67 + 40.

I think I’m getting closer to understanding Sfard.


Advice to Myself

Zach, on school-life balance:

If you had told me that it would take me 5 years of teaching to figure out how to mentally leave work at work then I might not have continued in this career. I’ve gotten incrementally better at it each year but this year I’ve committed to prioritizing it. Here’s a few things I’ve learned that help me do that. I hope you can, especially if you’re just starting out, find a piece of advice that will help you live a more balanced life.

and also:

Somewhere deep in the dredges of my brain I used to think that if I just spent as much time as possible thinking about how to fix the problems that arose at school (or in education in general) then I’d be able to fix them.

This doesn’t quite describe my situation. In my first few years I kept long hours, but it wasn’t from a perfectionist drive. The opposite, actually. I perceived (correctly) that I was doing an amateurish job in the classroom, and I wanted to throw hours at the problem (me). The hours would make me a better teacher, and I wanted my extra-miles to offset the trashy teaching I was offering the children.

I’m not sure if I have a message for my past. I’m pretty much the same person as I was in 2010, and I’d probably be inclined to react the same way.

I switched schools in 2013. My first school had a lot of problems, and it felt like there were a lot of big problems that were obviously outside of my control. (But you never can really be sure, can you?) And there’s no doubt that my move (to a school where things basically work) has reduced some of my pressure to put in extra hours.

I also had a kid.

But none of this is advice, either to my past-self or to others. So, let’s give advice.

To begin with, your life needs to be filled with something. You can’t simply stop thinking about the problems of teaching, children always win when it’s children versus the vacuum. You need to fill your life with something that is large enough to compete with your job. Possible candidates: religion, children, lovers, family, community, art, carpentry, any craft, any art, mathematics, reading.

Second, get off the computer in whatever way is possible. My two big victories this year: I bought a notebook for daily planning, and another notebook to keep track of when kids turn in assignments. I can’t say enough about how important it is to me to have highly visible, tangible artifacts for my organizational life. Calendars, planners, notebooks, folders. There’s no way around it: planning ahead is easier with physical objects.

“No ideas but in things.” If you’re working on teaching, something should come out of it. Spending an hour agonizing over a lesson while flipping back and forth to twitter is not work. Recognize that most of your evening and weekend work time looks like this, and then simply give up on this.

Conventional wisdom is that we live in a time of intense distraction, and that conventional wisdom is correct. That’s the most important thing to remember, that we can choose how to direct our attention by crafting our environments (rather than imposing our will on our environments).

From Dan:

I’ve grown to admire a kind of teacher I used to disregard – the teacher who knows she could create a better lesson than the one she taught last year, who knows she could help a student bring a B to a B+ with after-school tutoring, who knows she could do wonders coaching the basketball team, and who makes a principled choice not to do any of that.

If you want to make the calculation, make it this way: how can you be the best teacher for all of your students, both present and future? Are you doing your future students a service by being a hero in the present? Are you making this life unsustainable?

Final advice: do fewer things. The very minimum, if you can.

Against (a Certain Kind of) Public Debate


Perhaps I’m misreading Greg, but here’s what I take him to mean: there is a performative aspect for debates. “Performative,” in the sense that debate is for the benefit of others, “onlookers.” You’re not truly expecting to convince your opponent. Instead, you’re trying to influence the audience.

I don’t find this notion bizarre. It’s pretty much how all public debates work. Clinton isn’t hoping to sway Trump; Baldwin was speaking to the Cambridge students, not to Buckley. So all of that is fine.

Debate as a public performance is something I see a lot on twitter, and not just from advocates of traditionalism in education. It reminds me most strongly of the political left’s call-out culture, and I think a lot of what you might say about how traditionalist activists carry themselves on twitter has already been said about this brand of leftist activism.

The other thing about call-out activism — traiditional or not, in or out of education — is that it is absolutely exhausting. There is always someone wrong on the internet, the arguments are always the same, the sins are never new. I wonder if many of the gains in the last few years in face to this form of activism — again, in or out of education — are somewhat superficial, in that they aren’t much more than the mainstream trying to avoid this sort of exhausting, relentless critique.

But what’s the alternative?

Communities are more powerful than individuals, and can have more of an influence. This is a true truism. Beware activists that identify their struggle too closely with their movement’s. Not everyone has to be an activist, and not everyone has to be an advocate in precisely the same way, but if you’re saying that big changes are needed and all you can come up with is publicly critiquing people? Come on, build something.

The other alternative is to give up on online activism altogether. Not everyone must be an activist for what they believe. To be an activist is to adopt certain presentations and frames — among other things, advocacy demands presenting yourself as an expert.

Social media has had a lot of effects on our discourse, among them an enormous increase in the number of experts we have on any given topic.

I sometimes say something that is really easy to mock, which is that we should spend less time arguing and more time investigating. I don’t have much to say when people tell me that this is dumb, because it’s something that is a little bit ahead of what I’m able to clearly articulate. (That’s why I have a blog, though!)

Here’s why I think inquiry-over-activism is important. First, because it seems to me that honest, public inquiry’s stock is currently undervalued. Publicly critiquing ideas that you know to be obviously and badly wrong is a mode that’s being covered just fine, thanks. Second, because I think that publicly investigating and sharing an interest does far more to impact “onlookers” than yelling/calling-out. It’s engaging in a way that “debating for the benefit of onlookers” can’t ever hope to be. Third, because I think that there are moral risks to appointing oneself as a critic, or on activism that isn’t grounded in community. These risks are familiar, but real.


Thinking in this way has given me a simple test for deciding whether something is worth writing: Am I sharing something new that I discovered? If yes, then write and publish. If not, keep looking for something new to learn.

That’s the same standard I’ve found useful in deciding how to engage on twitter. Are we learning something new, right now? Am I understanding anything better through this debate? If so, engage. If not, attempt to change the course, but then punt or mute.


Update: My original title “Against (Most) Public Debates” was pretty bad, so I revised it.

There were two other things knocking around in my head as I wrote this post, and I wished I had shared them. Here they are.

From philosopher Mary Midgley:

What is wrong is a particular style of philosophising that results from encouraging a lot of clever young men to compete in winning arguments. These people then quickly build up a set of games out of simple oppositions and elaborate them until, in the end, nobody else can see what they are talking about. All this can go on until somebody from outside the circle finally explodes it by moving the conversation on to a quite different topic, after which the games are forgotten. Hobbes did this in the 1640s. Moore and Russell did it in the 1890s. And actually I think the time is about ripe for somebody to do it today. By contrast, in those wartime classes – which were small – men (conscientious objectors etc) were present as well as women, but they weren’t keen on arguing.

It was clear that we were all more interested in understanding this deeply puzzling world than in putting each other down.

I was also thinking of Lakoff’s explication of the “argument as war” metaphor:

Screenshot 2016-10-09 at 9.08.59 AM.png

But They’ve Never Heard of Positive Numbers

Ancient Greek mathematicians had no notion of negative — or positive — numbers. Similarly, for most young children, numbers are not signed. Children learn to count with natural numbers. At some point, they learn about zero. Later, they encounter (regular) fractions, decimals, and percentages. Typically, a child’s formal introduction to the notion of sign comes after all this experience. Interestingly, we have found that many students in the elementary grades have some familiarity with negative numbers but have never heard of positive numbers. These children inhabit intermediate worlds that consist of regular numbers and negative numbers before they begin to (intermittently) inhabit worlds of positively and negatively signed numbers.

Regular Numbers and Mathematical Worlds