Secret Squares and Stop Signs

I wrote a deeply sad post today. As I went to publish it, I hesitated.

Yesterday was my last day of school, and instead of ending with sadness, I’d like to end with a celebration — documenting joy!


The Power of Curiosity

This post is a tribute to the Power of Curiosity.

There are a lot of elements that go into great teaching. A quick google search yielded dozens of infographics showing the connection between content knowledge, and pedagogical knowledge, and pedagogical content knowledge…

One element I’d like to write about today is genuine curiosity. Genuine curiosity is infectious. It can spread through a classroom, sparking creativity, and encouraging students to dig deeper. I think this curiosity can start anywhere, but it’s especially helpful if the teacher cultivates it. As much as we may strive to position students as equal agents within the classroom community, we teachers have a lot of power and influence.

Genuine curiosity alone won’t create brilliant teaching, but it does have the potential to ignite.

Link to tweet

So here’s what happened when I was totally and incurably curious about student thinking — in one classroom, on one day, in one lesson.

Along for the Ride

Ideally, my classroom teacher colleagues and I plan together before a lesson. Of course, sometimes life happens, and we don’t get to plan the way we’d like. I show up, ready to go along for the ride.

Recently, I found myself along a truly beautiful ride. I ended up livetweeting it like a season finale.

It started in a third grade classroom. The classroom teacher loves teaching math. She’s licensed to teach it at the middle school level, and her content knowledge is immediately evident in the way that she poses probing questions.

“We’re about to start the 2D geometry unit,” the classroom teacher told me as I entered the room. “But I know kids often come in with lots of background knowledge, so I’m kind of curious to see what they already know.”

“Beautiful! I’m curious, too!” I told her.

Students organized themselves into pairs or triads, and began to sort a deck of shape cards.

The “Stop Sines”

I walked over to the first group. They had a few shapes — rectangles — positioned close to part of a pink sticky note that had been labeled “rectangles.” Fair.

What was to the immediate left of the rectangles is what piqued my curiosity.

The stop “sines”! I asked the students to explain: “These shapes are more ‘circlish’ than the other shapes,” all of which are quadrilaterals and triangles.

“You could write something on them, like STOP.”

Brilliant. They were looking at the shape holistically, but starting to name properties.

“Could you write ‘STOP’ on any other shapes?”

“Yeah, but if you write it on a rectangle it’s not like a stop sign.”

The “Daimonds”

I walked over to the next pair. “Tell me about your favorite grouping.”

“These are the diamonds. If you turn them they will make diamonds. Oh, wait, one of these needs to be moved. I think it’s a trapezoid. No matter how you tilt it, it’s not a diamond.”


At first, I was surprised to see the kite mixed in with the parallelograms. Maybe a key characteristic of diamonds is that opposite angles are congruent? As I excitedly attempted to draw more observations out of this pair, the Stop Sine duo called me over.

Staircases and “Trapizoids”

“These are the trapezoids. I wanted to call them staircases but Alice says trapezoid,” Jada stated crisply.

“Why did you want to call them staircases?”

“They all look like they have a side to walk up.”

“And why did you want to call them trapezoids?” I turned to Alice.

Alice paused. “They look like trapezoids to me.”

“Where have you seen a trapezoid before?”

“Okay, so you know like pattern blocks? The red one? That’s a trapezoid, and these kind of look like that.”

Jada continued: “Yeah, and they all have four sides. Oh!!! Maybe we should give the stop signs a new name!! They all have 5+ sides.”

Related to Hexagons

The classroom teacher and I giddily bounced around the room, listening to students share their thinking, and encouraging them with sweeping gestures, emphatic head nods, and exuberant yelps of “OH!”

A pair of students across the room independently created a group that matched the Stop Signs, but named it “Related to Hexagons.”

What made them related to heagons?

“The others have 4 sides or 3 sides. Oh! These are 5 or 6 or more sides! Lots of sides!” The student recorded “5+ sides” on the sticky note.

The “Sqarils”

Next to the Diamonds group, students had crafted a larger, more inclusive group of quadrilaterals that they named Sqarils.

“They’re all kind of like squares,” the student told me.

How are they like squares? “Four sides.”

Are they actually squares? “Just one. They others are like squares, but not really, so they’re sqarils.”

The Secret Squares

A nearby group heard me excitedly chatting with the Sqarils team, and wanted to share with me their own group of squares.

“Squares need to have all equal sides,” one of them boasted.

“That’s true!” I exclaimed.

They scanned the cards laid out on the desk. “But also, Ms. Laib, this one has four equal sides! So is it a secret square?”

“I love that! Secret square! What do you think?” I tossed the inquiry back to them.

Without hesitation, one of them called out: “let’s find more secret squares!!!”

I would never have been able to mask my enthusiasm for this new classification. Secret squares! Shapes that fit part of a definition, that push us to be precise in new and delicious ways!

What’s in a Name?

Students were creating definitions that sometimes matched the quadrilateral hierarchy that we teach, per the standards, and sometimes not. It looked like every single group in the class had a group labeled “diamonds” that included kites, which meant that there was something about diamond-ness that is separate from rhombus-ness.

This raised the question — how do we define a shape? Why do we create groups and categories and hierarchies? I mean, certainly it would be impractical to try to give a name to each and every possible triangle…

That which we call a rose
By any other name would smell as sweet. 

If we change the name, do some of the defining characteristics shift in translation? We sometimes teach kindergartners that a diamond — like the blue diamond in the pattern blocks — is really a rhombus, but I think every third grader in this class is operating from a different definition.

I wanted to know more and more about every student’s thinking. How could we dig deeper into how they were defining these groups?

Collect and Display

The classroom teacher glanced judiciously at all of the pink sticky note labels literally the students’ desk. She collected a few, and recorded them on the board.

“Here are some of the group names that you came up with as you were sorting,” she started. “What do you notice?”

There are some made up names. (The Sqarils)
There are some real names. (Rectangles)
There are some based on the number of sides. (5+ sides)

“Why might we group or name things based on their sides?” The classroom teacher asked, tapping her chin dramatically. I told you: she knows her stuff.

“It’s way easier to find something!”
“Our groups would be the same!”

I stood in the back of the class, listening to the conversation, completely enraptured. It may have taken one of the Secret Square students two tries to get my attention, and then:

“Psst! Ms. Laib, come over here! I found more secret squares!”

“So what makes these secret squares, and not just… square squares?” I asked.

“They have more than 4 sides. Except this one…” she wrinkled her nose, and pointed to the rhombus. “It’s got four equal sides but it’s not a square. It’s like… a tilty square. So it’s a secret.”


Meanwhile, the classroom teacher introduced the term quadrilaterals, stating that it would be the focus for the rest of the weke.

A pair towards the front of the room waved their hands excitedly. “We already have a big group of all the quadrilaterals! We just have to rename our sticky!”

“What did it say before?”

“All 4 sides. Now it says… quadrilaterals!” The student announced with a flourish. “Also cuatro is Spanish for 4 so this all makes sense.”

During the time I was speaking with the Cuatro girls, the rest of the class arranged their cards so that they had a stack of quadrilaterals.

“How could we sort these quadrilaterals?” the classroom teacher posed. She knew exactly where she wanted to nudge these students.

The Stop Signs Duo argued somewhat bitterly. One of them wanted to sort the quadrilaterals by perimeter, but the other lamented that “it’s going to take us forever to measure these shapes! There has to be something easier!”

A group towards the back of the room created a pile of “slanted shapes” and “not slanted shapes.”

While they had named these holistically, there’s something about the angles that makes some of these shapes appear slanted! I puzzled over how to get them to notice right angles even if they have not worked with the concept of “angles.”

What defines a rhombus? What defines success?

The classroom teacher chose to do this activity because it was an accessible, creative, and fun way to learn more about student thinking. This was a dynamic formative assessment, constantly shifting over the course of the lesson as student ideas shifted and took shape.

If I had to name some elements of success that contributed to this learning experience, I’d say we can thank:

  • The classroom teacher’s pedagogical knowledge
    She respected student thinking first and foremost, and gave them space to play with ideas in a mathematically creative way.
  • The classroom teacher’s content knowledge
    She nudged students towards ideas that we will formalize later in this unit, and helped them think about some big picture ideas around geometric classification.
  • The classroom teacher’s pedagogical content knowledge
    Pedagogical content knowledge is, to be reductive, how teachers relate their content knowledge and their knowledge of pedagogy to adapt to specific mathematical situations.
  • The classroom teacher’s genuine curiosity.
    We wanted to know what students were thinking!

Our genuine curiosity led us to celebrate students mathematical ideas, and cheer when students revised them and defined them. It allowed us to dig deeper into concepts. It was a vehicle for precision in both mathematical language and mathematical content. The decision to honor student thinking and then nudge it forward opened students to greater mathematical agency and — I hope! — stronger mathematical identity.

Stop Signs No More

Our love for their ideas did not mean that we would have them categorize regular polygons with >4 sides as “stop signs” forever. The students would continue to knead their understanding of quadrilaterals over the 2D geometry unit, shaping concepts until we started to recognize patterns in how we name things: the number of sides is important, as are the relationship between those sides and also the size and relationship of angles.

It would have been overwhelming to deal with all of the names and categories on the very first day, without having experience with the figures. And it would be negligent if we willfully held students back from content knowledge — content knowledge that we have, and that our state standards expect them to learn. And within all of that, there’s space to appreciate Secret Squares, both as a noun (its own entity) and a verb (a way to flex our geometric muscles).

There is so much power in honoring what students have to offer, and then helping them to grow.


  1. Oh my goodness, my mind boggles, but I am actually completely enthralled with excitement about the learning going on. Each shape is looked at so carefully for future lessons, creativity shines through, cooperate learning galore, with and from each other, teachers and students alike. Inspiring learning in capital letters. These students have so much to build upon. An absolute ‘WOW’ time was had by all.

    Liked by 1 person

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