“I’m sorry, we’re only loosely doing math today,” Jessica apologized as I walked into her kindergarten classroom. “We’re deep into our structures unit in science, and I’d like to keep going.”

Yes!

I love the structures unit in kindergarten. There are so many opportunities for students to play with shapes, devise and revise plans, and make mathematical connections. Mathematical thinking is not confined to the “math block” on a teacher’s schedule. Of course, it helps for a teacher to look with a mathematical lens, to ask questions that elicit mathematical thinking, and help students attend to their own mathematical observations about the world.

Yesterday, groups of students were given a photograph of a structure — from the Petronas Towers in Kuala Lumpur, to the Eiffel Tower, to an elaborate tree house and a water tower.  They were also given building materials like Magnatiles, foam blocks, Cuisenaire rods, and straws. Students were then tasked with building a replica of their structure using their given building material.

Jessica had already posted photographs of yesterday’s structures on the bulletin board in the hallway. Students had clearly done a lot of mathematical thinking! Look at this group’s replica of the water tower. They attended to those long “legs” of the power, and build them with a square base. These legs supported a round basin, which they recreated with beautifully circular frames. There was a lovely interplay between the angular and round shapes, coming together to form a deceptively simple design.

Another group built a large castle, with 3 turrets that stretched higher than the surrounding wall. They were attending to quantities, to size, to shape. Even from a 2 dimensional image, the students knew that they wanted the surrounding wall to be rectangular.

Today, students would be given the opportunity to create a replica of the structure they worked with yesterday — but with new building materials. Some materials were easy to use for construction, like these foam blocks. These students recreated a castle with three visible turrets, and a large surrounding wall.

Some materials more challenging, like the Cuisenaire rods, which students struggled to keep upright.

I sat down with a group working on the Qutb Minar complex, a UNESCO World Heritage site in Delhi, India that contains both the minaret and the Quwwat-Ul-Islam Mosque.

Yesterday, they had used Magnatiles to build their structure. “It was easy to build. The blocks click,” Makayla explained.

“I made the tower,” Vihaan stated proudly, as he traced the edges.

Today, they were given the straws and links tool that the water tower group had used the day before. The three students sat down together to make sense of the materials, each fiddling with the connectors and the straws until they finally pushed together. I assumed that Makayla, shown here on the left, was making the round base of the tower, and that Eleanor, who was starting with the square base in the upper right corner of the image, was working on rectangular mosque, but it turns out they were building two different interpretations of the same part of the structure.

“The building is round!” Makayla insisted, pointing to the dome of the mosque. “So I’m making a circle!”

“It’s flat!” Eleanor shot back. “It doesn’t need a circle!”

Vihaan shrugged, as he continued to build a long, straight line of straws. “It’s the tower.”

As an adult observer, this felt like such a clear peanut-butter-and-chocolate moment. The circle and the rectangle go together! The base is rectangular, while the dome is a round hemisphere! But the girls couldn’t see it.

“Hmm. How can we make these parts into one structure?”

“I’m making the tower!” Vihaan reminded them.

The girls frowned at one another’s shapes.

“Where do you see something circular?” I asked Eleanor, who was mostly ignoring me to build up her rectangular base.

“It’s mostly a rectangle, so I think it should be a rectangle.”

“Do you see anything that reminds you of a rectangle?” I asked Makayla, who was sheltering her circular base with a protective hand.

“I guess. But the circle is important!”

Creating a replica is all about extracting the important thing. It reminded me of “The Important Book.” The Important Thing about building the Qutb Minar is that it has a tall tower, that it has a small building with flat sides, and that the small building has a round dome top.

I pulled out my phone to examine the kindergarten standards for geometry.

K.G.B.5
Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.

Well! We were certainly doing that!

Then I saw this:

K.G.B.4
Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/”corners”) and other attributes (e.g., having sides of equal length).

I wanted to hear students talking about the shapes — what the sides should look like, what the corners should look like, how many sides they need to build, what the shape of the base is, etc. There is also a standard (K.G.1) about describing relative position, which seemed like it might be helpful as students join different constructions.

Makayla continued to build, adding a few straws to her base to make it a larger circle. “It’s too hard to put together,” she lamented of the smaller circumference, which kept breaking under the stress.

Meanwhile, Vihaan and Eleanor teamed up forces. Vihaan released that his long, thin rope of straws — the important thing about the tower is that it has long edges! — might work well in conjunction with Eleanor’s square base.

“Let’s make it longer.”

“No, taller.”

“It needs to be straight.”

“It needs a triangle on top.”

Soon, they had build a tall tower. Meanwhile, Makayla was struggling with her circular base. The straws popped out of the connectors every time she tried to add another layer.

At left is their final product. The had a big tower to the left, like in the picture.

“It’s really tall.”

“It’s big.”

“It’s straight.”

“It has triangles.” Any other shapes? “It has squares.”

The smaller building, the masjid, was harder for them to describe.

“It’s shorter.”

“It’s attached to the tower.” So it’s next to it? “Yes, it is next to it.”

What shapes do you see? “I don’t know. It’s supposed to be like a circle.”

I asked the students whether the magnatiles or the straws were better building materials, and they all agreed that while the Magnatiles were easier to build with, the plastic straw structure looks like more like the structure in their picture.

Meanwhile, across the room, three students were working to build a replica of this structure. (It is the Broadway Tower inWorcestershire, England, the second highest point in the Cotswolds.)

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The Important Thing about this tower is that it has 3 turrets, and a balcony (see on the right side of the structure, close to the photograph). Originally, the central turret had an extra 15 magnatile squares stacked up in its base. As an adult, I assume that the turrets are actually of equal height, but the students do not know that. I did ask them about the height.

“These two are the same. This one is taller.”

How much taller? “A little.”

How about in your structure? Oh. Hmm…. How can you change it to match the photograph better? We can take some squares out of the middle! Then it will be shorter!

I was impressed with how student were analyzing their shapes, and attending to the geometric features in their structure.

I saw Jessica across the room. She was asking a group about their treehouse made out of cuisenaire rods. She probed their thinking about whether to make their replica flat against the table, or build it up, nicely hitting common core standard K.G.3 without even realizing it. She asked them about some of the important features, and what shape she made. She listened intently to students, with her head tilted and her brow furrowed.

After the lesson, Jessica and I debriefed briefly about all of the mathematical ideas present in the lesson — some inherent, and some because she drew them out. She made beautiful connections that helps students improve their mathematical observations and thinking about their structures.

I looked up the geometry standards once again, and sent Jessica a follow up e-mail with some additional lines of questions — although her intuition was already spot on.

As teachers, we can model how to mathematize the world. (Dan Meyer (@ddmeyer) has been writing about this lately both on twitter and his blog.) Some of the thinking here is explicitly about content (e.g. what shape is the base of the structure) and some of it was about important avenues of mathematical thinking, like the process of revising our work.

What is the value in mathematizing outside of “math class”? When do we draw attention to these mathematical ideas, and how do we improve the clarity of our connections?