Yesterday, I was the lucky duck working with a team of three teachers who each teach multi-age 3rd/4th grade. We were all in one class together in the morning, and then got to meet during their common planning time later that day.

The kids have been working on perimeter, mostly of rectangles but also of other polygons. The first problem yesterday emphasized decomposing a 14-sided polygon into smaller rectangles in order to find the lengths of all the sides and then calculating the perimeter. Rectangles were in the air.

The second problem was this one from *Investigations*:

“The perimeter of Pilar’s yard is 100 feet. Draw a picture of what her yard might look like, and label each side.”

Unsurprisingly, we mostly got rectangles from kids, and mostly these two variations:

There was a little discussion at tables about whether this 25 x 25 square is actually a rectangle. Lots of kids have the classic misconceptions there. Oh how I wish we called squares “square rectangles” or “equilateral rectangles.” Sigh.

One student made a pentagon instead, which I totally loved.

And then one student, pseudonym Jesse, made this shape:

Look closely. I noticed the dimension on the left-hand side is 0 yards. So are some of the ones on the right, but not all. I also noticed the long line across the top is 30 yards, and the series of short line segments on the right hand side that are all about the same length start with labels of 30 yards, 6 yards, 1 yard.

After the gallery walk, I headed over to Jesse. He made me laugh because he said, “I could see you squinting at my board from across the room!” I told him how cool it was that he took a risk and tried out a different shape, and I asked him how he did it.

“I made a cool shape, and then I started putting numbers down. But the problem is I used up my 100 and I still had a lot to go, so I made some of the lengths zero.”

“Interesting. I’m wondering what would happen if you tried to build this yard?”

Jesse said, “You couldn’t. It wouldn’t work. The zeros would mess you up.”

I asked, “Do you think you could make a yard

this shape,thatdoeshave a perimeter of 100 yards, that you could actually build?”Jesse said, “Yeah, but I’d need to erase all the numbers and start again. I’d try smaller numbers first so I wouldn’t run out of fence.”

We had to stop talking then, but I grabbed his board to discuss with the team later at common planning time. The kids had moved on to talking about the third problem they’d worked on, which was to draw three different rectangles that all had a perimeter of 20 units. The teacher had them post different rectangles on the whiteboard, and we saw all the whole number solutions: 1 x 9, 2 x 8, 3 x 7, 4 x 6, 5 x 5. The teacher asked a money question:

“Do we have all of them?”

Love that.

One student, pseudonym Zeke, said “No. There are lots more. Like, I could make one that’s 15, 2, 2, and 1. They add up to 20 also.” He held up a piece of paper that had a mass of zigging and zagging and curving lines and pointed to it as evidence.

This was the most interesting moment of the math class. Both the classroom teacher and the paraprofessional physically moved toward Zeke, lowered their voices and started talking at the same time:

“No, you can’t because…”

“Remember, you have to make a rectangle and rectangles have…”

In keeping with the norms we teachers have as a team of learners, I paused my colleagues for a second and asked if kids could discuss Zeke’s claim instead? Thumbs up. I asked the students whether they agreed or disagreed with Zeke’s statement that a rectangle could have side lengths of 15, 2, 2, and 1?

Students discussed the claim at their table groups. As we listened in, we heard a few useful arguments pop up. Some tables talked about both pairs of sides of rectangles needing to have equal length. Other tables talked about whether you could make a closed shape with those dimensions. Other tables were full of kids trying to draw a solution to his challenge.

We also noticed, though, that several students said you *could* make a rectangle with those dimensions. They had different, confusing arguments, and they weren’t leaning on the properties of rectangles at all.

And we noticed that no group had put together all the pieces. There might be one student talking about opposite sides having equal length, but they weren’t putting that property together with other properties of rectangles in a coherent way.

Time was up and we left it there.

At the end of the day, we were so glad to meet at common planning time. The classroom teacher thanked me for stopping her from correcting Zeke, and thought his claim made for a worthy discussion. We talked a bit about this—about how to view mistakes as opportunities in the classroom, and to free ourselves of the pressure to correct kids right away. We talked about not worrying that posting or sharing wrong answers will confuse kids. We talked about this lovely feature of mathematics, which is that we don’t have to jump in as the authority figures of what students “can” or “can’t” do; rather, we can trust that, in almost all cases, students can determine the truth for themselves by delving into the mathematics. This quote from *Making Sense *captures this idea beautifully:

“In traditional systems of instruction, teachers are asked to provide feedback on students’ responses, to tell them whether or not they are right…this is almost always unnecessary and usually inappropriate. Mathematics is a unique subject because…correctness is not a matter of opinion; it is build into the logic and structure of the subject…There is no need for the teacher to have the final word on correctness. The final word is provided by the logic of the subject and the students’ explanations and justifications that are built on this logic” (Hiebert et al. 1997, 40).

We also talked about the importance of non-examples, which made me think of this perfectly put tweet from Kate.

I told the teachers that, as a 4th grade teacher, I always had to teach kids what rectangles are. Kids pick up the crummy board-book definition of rectangles as shapes with 2 long sides and 2 short sides and square corners. (All the more reason why I can’t wait to publish Christopher Danielson’s better shapes book next year!) As teachers, we need to create conditions for kids to add depth and sophistication to their understanding about shapes. I thought Zeke’s claim was a fantastic opportunity for students to think about what a rectangle is and isn’t by focusing on the *properties *of rectangles, not just visual arguments.

The four of us kicked ideas around for where to go next. We had this open question: could you make a rectangle with side lengths of 15, 2, 2, 1? We all felt like the kids should take that question up by trying to build the rectangle. We settled on this plan for today:

The teacher would give each student 20 toothpicks and define each toothpick as one unit long. She’d ask students to make all the rectangles they could using all 20 toothpicks, without breaking toothpicks. They’d synthesize their list of rectangles to see if they had all the whole number solutions. Then they would take up Zeke’s challenge. Could they use their toothpicks to construct a rectangle with side lengths 15, 2, 2, 1? The teacher would use this exercise to draw out the definition of a rectangle, focusing on properties.

I wish I were going to be there today so I could see how it goes!

The four of us moved on in our planning time, and I showed them Jesse’s diagram for the 100-foot perimeter problem. This one.

Nobody had noticed it during the lesson, and we had a great conversation about it. And then the classroom teacher laid a whopper of a question on me:

“Do you think I should take up this work with the class?”

Holy smokes. So much ran through my head right then. Mostly, I was thrilled, overjoyed, elated by her question. Remember, earlier that day she had started to correct Zeke instead of consider his claim. I had just spent all this time trying to help this group understand the value of a piece of student work that reveals mathematical misunderstandings, or provides an opening to dig into a rich piece of mathematics. She was obviously listening and learning!

Once we decide to teach this way, though, the questions get harder, not easier. If we’re going to be *responsive* to what our students do and say in mathematics, how do we decide what to respond to? What makes for a worthy mistake to take up as a whole class? How do you know? By what criteria? We obviously can’t launch deep investigations into every single piece of student work, so how do we decide which ones to pursue?

This is one of my favorite things to think about. There’s a penciled-in plan for me to write a second book after I finish *Becoming the Math Teacher You Wish You’d Had. *The working title in my mind has always been *Teaching from Mistake to Mistake. *I want to get into these thorny questions of how we respond to students’ mistakes, what mathematics lies underneath different mistakes, what mathematical goals we have for our kids, how we decide which mistakes to take up in a big way, and how we decide which mistakes *not* to take up with the whole class. These kinds of instructional decisions are the ones that keep me learning and growing as a teacher. They’re also the kinds of complex instructional decisions politicians with simple answers and writers of scripted curriculum DO. NOT. GET.

So, as a group, we talked about what mathematical issues would emerge if we took up Zeke’s claim or if we took up Jesse’s work. My counsel ended up being that more students would benefit from the exploration of Zeke’s claim because the kids really need to investigate the properties and attributes of a rectangle. In my conversation with Jesse, it became clear to me that he wanted to head to a place of guessing and checking different perimeter lengths to see if he could sum to 100. There’s lots of fun math in that, but I think we’d end up investigating addition more than geometry, and the teacher’s goal is a deep investigation of geometry right now. I also think Zeke’s claim is *accessible to every student* in the class, and *all* students would benefit from exploring it. We talked about giving Jesse time to try to solve the problem he’d set out for himself because there’s powerful learning he can do. Of course, if his work spreads so other kids are trying to solve such complex problems during their next rainy-day recess, awesome. But Jesse’s diagram is not nearly as accessible to all kids, and I’m not convinced the time it would take for all students to make sense of it would have enough of a mathematical payoff to justify that use of class time.

It’s not a clear-cut call, though. I have mathematical reasons to say class time would be better spent testing Zeke’s claim rather than trying to make sense of Jesse’s work. But am I considering the right factors? Is it the same call you would make? How do we answer this question: which mistakes are worth pursuing as a whole class? I hope you’ll give me some pushback and ideas to think about in the comments.

*Update*

The teacher wrote me and told me how the toothpicks exploration went:

“The toothpicks worked great! I encouraged them to move them around and see if they could make Zeke’s suggested rectangle measurements work. It didn’t take them long to realize there was no way to make it work and most kids said for one reason or another. Yet I pushed them further to list ALL the reasons that it could not be a rectangle, based on what they have previously learned about rectangles. It was a really good math discussion to not only clarify all the characteristics that make a shape a rectangle, but to demonstrate how you need to prove it with facts – plural! That the argument one way or another gets stronger with more facts.”

References:

Hiebert, James, Thomas P. Carpenter, Elizabeth Fennema, Karen C. Fuson, Diana Wearne, Hanlie Murray, Alwyn Oliver, and Piet Human. 1997. *Making Sense: Teaching and Learning Mathematics with Understanding.* Portsmouth, NH: Heinemann.

What a rich classroom and teacher discussion scenario! Thank you Tracy. Though I am not in the classroom, you painted a very clear picture. Your reasons for focusing on Zeke’s claim rather than Jesse’s strike me as very sound. You are considering the mathematical goals for the class as a whole, what is accessible to them, and what seemed to bring up confusiions for many. Your knowledge about the depth of understanding students need to develop about rectangles also helped you make the decision. I applaud the way the teachers engaged in thinking through next steps. Sounds like they are really deepening their understanding of pedagogy.

Thanks so much for sharing!

Nancy Belkov

Thanks, Nancy!

Tracy, Glad to see you back on line. I agree with Nancy Belkov that the reasons for focusing on Zeke’s claim create a more accessible problem and get to the properties of a rectangle which is a fundamental understanding. I really appreciated this glimpse into your work and the teachers you work with. I’m going to share with my colleagues as an example of a teacher “mistake” and the learning that comes from those mistakes, too.

aloha, Becca

I hope you’ll keep me posted, Becca. I’m so curious where that conversation will go.

This looks like some understanding was achieved. I like it. I liked the toothpicks. Since the perimeter is fixed have you considered using a loop of string as well?

Thanks for your comment, Howard. We did consider string, but decided that the difficulty of making truly straight lines with string would make it harder to think about that property of rectangles, especially because Zeke had curved lines in his drawing. And then there’s the issue of making vertices. So we toyed with it, but decided the toothpicks would help us accomplish more goals.

I did dump the idea of sticking thumbtacks into the interactive whiteboards (for various reasons!), and then thought that the students could use their fingers. For big loops two people needed. The rounded corners are a mild distraction, but the general idea of what a rectangle looks like could come to life.

This looks like a good idea to me (although I know they’ve already got a good plan and they should go with that) – geoboards would do this well. Here though, the students would have to count their perimeters, which could prove interesting with diagonal rectangles!

Ruler ?

Hi Tracy! Fun to read this. Now I want to solve Jesse’s diagram… The toothpicks seem like a good idea in that they aren’t as fungible as hand-drawn lines are. Two lengths is two lengths, you can’t draw another line the same length and call it 4. I guess the interesting part of Jesse’s would be in coming up with numbers that approximate the right scale (the right side having to equal the left, in increments plausible to what he has drawn). But that does seem like more an individual problem that would be fun to work out, maybe not suited to group work. From a non-teacher perspective. 🙂 Also, interested to know whether it’s just coincidence that the two examples of kids thinking outside the rectangle box (ha ha) were boys?

I like that word, fungible!

Reminds me of Edward Lear. “Runcible” needs a mathematical usage.

Me too! No surprise. Emily is one of the best sources of great words I’ve ever known.

Oh my goodness, Em, I love seeing you here! And if you hadn’t called yourself out, nobody would know you’re not a teacher. You’re asking great, teacherly questions! I thought about the gender issue too. It’s a consistent theme that boys are more encouraged than girls to think outside the box mathematically. Maybe it will make us both feel better to think about that the pentagon originator was a girl? It was!

(Loved your email. Can’t wait to write you back. Or maybe we should GHO or skype? Lots to catch up on. xo T)

I love that – “In keeping with the norms we teachers have as a team of learners, I paused my colleagues for a second and asked if kids could discuss Zeke’s claim instead?” To be able to pause colleagues without it being felt as a reproof speaks of a respectful and trusting relationship among you, and a concern for getting better as teachers in place of any defensiveness.

It strikes me that these mistakes, when we can run with them as provocations for further learning, not only have an important cognitive function, they work narratively too. They set up a tension which focuses our attention. Frodo’s journey to Mount Doom would not have been as unputdownable without Gollum, the Round Table needed the sharp-tongued Sir Kay to provoke the other knights and show their true courtliness. So long as the student who made the mistake is made to feel comfortable with this, it can energise a lesson.

And of course they often push the limits. Which is the spirit in which we do mathematics. What if we could have numbers on the other side of zero? Or more than three dimensions?

It’s great to hear the title of your second book Tracy! What a great project!

I was looking at that L-shape of mine again

http://y4ist.blogspot.fr/2015/01/three-awful-awesome-mistakes.html

with teachers, discussing how in valuing talk we must also make it OK to make mistakes, and get used to seeing the ideas that are inside those mistakes. I think there’s definitely an understanding to be built up here which will change our teaching!

Thank you, Simon. I’ve loved all your comments and tweets today.

As for that climate, all props go to Elham Kazemi, Allison Hintz, Lyndsey Gibbons, and Teresa Lind for teaching me to think about how to create a true community of learners and join it as a member. I’ve been watching Elham position herself as a learner alongside teachers for years, and it’s just beautiful. Last year at NCSM/NCTM in Boston, the team showed videos that really drove home how essential that culture is, and I’ve been working on it ever since. The only way this coaching thing works is if we all learn together.

Have you watched Elham’s #shadowcon15 talk, “Sitting Among Students?” It will give you a sense. http://www.shadowmathcon.com/elham-kazemi/

Tracy

Yes! I’ve watched Elham’s talk at least three or four times! And recommended it to colleagues. Truth is though, we’re just at the beginning of doing this kind of thing, and coming from a UK-type experience where observation is most frequently a top-down evaluative event.

We’ve just had a conference at our school where teachers have shared the things they really want to share, and my sessions were co-presented, so we had a good process of visiting each other. We’re moving to a new curriculum (The IB PYP programme) and it will hopefully be a chance to establish a collaborative learning atmosphere where visiting each other is part of the learning, and learning is so central that we’re comfortable with making and receiving suggestions. It’s great to have models to point to!

Sounds like you’re making great changes, Simon. Keep us all posted!

Hi Tracy,

Super great read and just what I needed to be reminded of! I have been struggling as I modeled number talks in over 50 classrooms over the last two weeks about how to handle all of the misconceptions that come up with students…the way you brought it all back to the objective and thinking about which mistake would produce the most mathematical growth related to the objective was spot on and something I wish I’d asked teachers, and read two weeks ago! Also, I think I’m going to try to snap more evidence of student thinking so when I meet with teachers I can ask them if there is one they’d like to cover in the future, or at least reflect on when we’re together! Thanks for putting down your ideas and experiences so we can have a window into what you’re doing to help teachers grow!!! I’m continuously inspired by you amazing math folks!!!

Did you say 50 classrooms in 2 weeks??? I can’t get past that number. That must be amazing and a whirlwind at the same time. I’d love to hear more.

So glad the blog was helpful. I was realizing today I didn’t write the word I actually said in our discussion, which was *fruitful.* We talked about thinking about mistakes that would lead to more fruitful explorations/conversations/etc. I have a lot more to think about if I’m going to try to nail down everything that comes into that decision. I mean, I didn’t even touch social goals for specific kids, class dynamics, or highlighting mathematical habits of mind/SMPs. Teaching is so complex!!

Tracy