Building Understanding in Two Dimensions

Thanks to funding from our superintendent, Dr. Flores, we were able to purchase a brand new geometry tool for all of our elementary campuses – Exploragons. They’re just little strips of colored plastic that snap together. So simple and yet chock full of so much rich mathematical possibility!

Today we invited 3rd grade teachers to a half-day professional development session to do some playing, exploring, and analyzing with this exciting new tool. (Thank you again to Dr. Flores for covering the cost of all the subs!)

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We started the session with a challenge inspired by an activity in Van de Walle’s Teaching Student-Centered Mathematics, Grades 3-5. We chose this activity on purpose as a provocation to get the teachers invested in the session. (Thanks to @mathinyourfeet for introducing us to the term provocation in our discussions at NCTM Annual last week!)


We asked the participants to sketch their prediction on a piece of paper and name the figure if they could. Then we asked them to share out their predictions. Next, we showed them how we could use the strips to mark off the vertices and then use a ruler to connect them to form a closed figure.

We compared the final result with their predictions and came to the conclusion that this figure can be called a:

  • polygon
  • quadrilateral
  • parallelogram
  • rhombus

We quickly talked about the first three names, but on the rhombus we challenged them a bit.

“How do you know this is a rhombus?” They of course said you could measure the sides. “You didn’t need a ruler to tell me you thought it was a rhombus, so how can you prove that it is one without a ruler?”

This made for a fascinating discussion involving diagonals, halves, and right triangles. We don’t want to give away the answer, so we’ll just leave you with that. In the end, we all were convinced it is a rhombus without having to touch a ruler.

And as it turns out, we decided that of all the possible names to call this shape, rhombus is the most precise because it accounts for the most properties of the shape.

Lots of great work and discussion so far, but this was all really a purposeful lead in to our next activity: the design challenge.


We gave the teachers enough time to create at least three shapes. When everyone was done we asked them to pick their favorite shape that they wanted to share with the group. In order to model how teachers might scaffold an activity like this with their students, we brainstormed geometric names for their shapes and properties that they might use to describe them.


Then we gave participants a minute to label their chosen shape before we took a gallery walk around the room so they could see what shapes everyone else was showing off.

We came back together to debrief:

  • What different polygons did you see during the gallery walk?
  • What did you learn about polygons and/or diagonals today?
  • If you could keep exploring diagonals with all of the Exploragons (not just 1 yellow and 1 orange) what question(s) might you want to explore?

A few great follow up exploration questions came out of this discussion, including:

  • We made triangles and quadrilaterals. Is it possible to make other kinds of polygons such as pentagons?
  • We didn’t see anyone make a square or rectangle. Why is that?

Sadly, we didn’t get to continue exploring these questions. With their minds sufficiently provoked, it was time to move on to some bigger ideas.


In Teaching Student-Centered Mathematics, Grades 3-5, Van de Walle talks about two goals of geometry instruction, spatial sense and geometric content. The geometric content side we feel like teachers are already familiar with, that’s our state standards, but we weren’t sure they were as comfortable with the spatial sense side of things.

We gave half the teachers a short passage about spatial sense from Van de Walle’s book and the other half the grade 3 geometry TEKS. After giving them a few minutes to read and make notes, the teachers partnered up to share observations and tackle the big question about how these two goals work together to help students grow in their understanding of geometry. We can’t say that we came up with the answer, but we felt like we succeeded in our PD goal which was to open the teachers up to the idea that there are two goals of their geometry instruction instead of just the one, teaching their content standards.

This conversation led into an introduction to the Van Hiele levels of geometric thought.

Planning and facilitating this conversation actually helped us better understand the way the Texas standards (TEKS) do nudge students along from level 0 in Kindergarten to the cusp of level 2 by the end of grade 5. However, it’s the geometric experiences we plan for our students that are going to matter most in actualizing what our content standards progression offers. Teaching the content standards isn’t sufficient if we aren’t creating numerous opportunities for children to explore, talk about, and interact with these geometric ideas.

We closed the discussion with this quote.


The role of students is to play and explore relationships, but teachers have a role as well. We need to guide students to learn how to do these things in a way that helps their geometric thinking deepen and become meaningful. Its similar to an issue our curriculum counterparts in science faced.

In the past they faced the problem that students did lots and lots of hands-on science activities, but they weren’t necessarily walking away with concrete and actionable science understandings. Science concepts are not magically imparted by mixing chemicals or watching butterflies emerge from cocoons. They came to the understanding that it’s the role of the teacher to create opportunities for students to debrief, reflect, and consolidate the learning that they’re doing through investigations and other hands-on activities.

Since Van de Walle emphasized play in his quote, we decided now was a perfect opportunity to do just that.


Up until now, the teachers had only had their hands on one yellow and one orange Exploragon strip. Before diving into any more structured activities, we gave them 10 minutes or so to let loose and have fun with entire bags of them. It was so much fun to see what they came up with!

After playing for a bit and admiring their creations, we came back together for one final question to close out this section of the session.


It was encouraging to hear the teachers already noticing a variety of ways that the Exploragons could support their students’ learning back in the classroom. Unlike straws and pipe cleaners – as fun as those can be to build with! – the Exploragons are beneficial because the pieces are uniform sizes. This allows for some very precise exploration and discussion. If you use four red strips, for example, you’ll know you’re using four equal-size lengths.

The teachers also loved how dynamic the shapes are. Triangles tend to be one and done, but other shapes can be tilted, slanted, or reformed in a variety of ways. This creates some exciting exploration and discussion possibilities!

Which is exactly where we went to next in the session. For the remaining hour and a half or so, we tried out a variety of different activities that the teachers can take back and do tomorrow. (And after the session, we wouldn’t be surprised if every teacher in the session headed straight to their campus library to check them out!)

We’re going to close out this post here and hopefully write up separate posts that go into more detail about the activities we shared. It’s one thing to see potential in a tool and another to get specific examples of how you might use it to advance students’ geometric thinking.

Thank you again to our superintendent, Dr. Flores for making this all possible. We’re so excited to get these tools into the hands of our RRISD elementary students!


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