It all started with the tweet from Sarah Caban a few days ago.


I decided to have a chat about it with the Kindergarten class, ask my grade 3 students, and one of our grade 5 teachers kindly agreed to ask the same question to her grade 5s. Here are the answers and some context from three different grades. At first, I planned to throw it all into a google doc and share with Sarah. Kindergarten kids’ ideas got me excited. When I started reading the other grades’ responses, I realized it’s worth sharing with anyone who might be interested.


Students in kindergarten have already learned some shapes vocabulary and identified different shapes in their environment. I started with pointing at their shapes wall, “I see you have been learning about shapes, we are learning about shapes in grade 3 too. Can you tell me what does it mean – “a shape?” A forest of hands.

“It’s a heart!”

“It’s a diamond!”

“It can be a circle.”

“A small triangle can be a shape.”

Students were pointing at shapes around them.

After recording all their “shapes” ideas, I asked, “What do all these things have in common? Why do we call them all shapes?” I was surprised how quickly kids considered the question and were all ready to discuss their ideas.

“Shapes are everywhere”.

“Everything is a shape but I don’t know what shape is.”

“We are made out of shapes”

Then someone mentioned that some letters are shapes. But not all. Students did not seem to have a consensus on that point, and the discussion continued around particular letters.

“C is not a shape, but you can make it a shape.”

Our kindergarten teacher who kindly invited me to talk math with her kids today, ask the student to explain what he means.
“You need to draw a straight line to make it a shape.” He drew a line connecting two “ends” of the letter C together. “Now it is a shape.”

Grade 3

Grade 3 students have just started talking about polygons and sorted their polygons and non-polygons into two groups. Their thoughts about shapes were likely affected by this recent activity. Students answered the question on the post-it notes and we did not have time to discuss it yet.

“A shape is something that has vertices and edges.”

“A shape is random line but with curves and edges.”

“A shape is a structure that most things are made out of. It is something that makes up everything.”

“A shape is something that you can outline.”

“Shape is something that you can use for a pattern.”

“A shape is something that connects. A shape means a thing that has no opening.”

“Shape is an object.”

“A shape is something to represent slots, key slots and other items.”

“Shape is 2D which means it’s flat.”

“A shape is something that is 2D or 3D and it needs a face.”

“A shape is something with vertices and everything is a shape.”

“A shape is something that has to have vertices/faces/edges and more than 2 edges or vertices.”


And the one that really made me pause and contemplate.

A shape is a 2D or 3D object that is drawn or held by humans.” – I thought bringing in the point that shape is something manipulated by people was really interesting.

I am also curious about this.

“Shapes are everything you see in life. You can find anything and it falls under 1D, 2D, 3D and 4D shapes.”

Grade 5

Grade 5 students worked on describing 3D and 2D objects and sorting quadrilaterals in the beginning of the year. They explored points, lines and did a lot of “hands on” geometry work.

“A shape is a thing that makes everything.”

“A shape can be a polygon and it can be irregular.”

“A shape is some sort of structure.”

“A shape is a structure of an object with all connected lines. Sometimes shapes have lines with holes.”

“A shape is a form of geometry that represents objects in their form. They are the appearance of an irregular or regular object.”

“A shape is something that can have sides, vertices, straight edges or none, so they can be anything like a soccer ball.”

“A shape is something like a symbol of math.”

“A shape is geometric combination of lines and curves making a closed region.”

“A shape is a closed 2D object with nothing inside it.”


I found it interesting to observe how the intuitions unfold with age, to see how students would use new and more sophisticated vocabulary to grasp the same concept. How they start looking at new mathematical concepts and to compare them with their idea of “shapes”. How they try and test new learned attributes to define a shape. When I started organizing these notes, I realized how much this simple question can actually tell me about my students’ mathematical thinking.


When other teachers look at all this students’ thinking, what do they notice? Just like these kids use different structures to define “a shape”, how are our “teaching structures” that we use to interpret students’ thinking are the same or different. Not sure it makes sense. “Everything is a shape but I don’t know what shape is.”


5 thoughts on “Everything Is a Shape But I Don’t Know What Shape Is.

  1. I love this post. It leaves me wondering so many things! You mentioned intuition and that is what I was wondering about the whole time I was reading. The description that spoke to me the most – the one that I identified with the most was from a Kindergarten student – “Everything is a shape, but I don’t know what shape is?”. This is what I was thinking when I originally asked the question. “Shape” isn’t a math term, but our collective understanding of it- as a concept- encompasses so much math thinking. I can’t help but wonder what would have happened if we had asked the same question about the word “polygon”. I suspect the answers would be more consistent, but less creative?? What does that mean for us as mathematicians and math facilitators? It reminds me of T. Zager’s book. I just finished the chapter that discusses how to balance honoring mistakes with teaching precision. I wonder if the same thing applies here – how do we honor precise vocabulary with intuition and ownership of ideas? I kind of want to argue that the K student is the most intuitively connected to math. That is probably a stretch because some of the 3rd and 5th students answers are also intuitive. One said “A shape is a thing that makes everything.” Maybe I am not using right the word to describe what I am noticing – maybe it isn’t intuition – maybe it is wonder. The K student was the only one who had a question in her/his answer, I think? That is fascinating to me because it is where I am now. As an adult, I am trying to strip myself of my formal relationship with math – not the formality – or precision – of the subject itself, but the formality of my relationship with it. I don’t want to have such a formal relationship with math anymore. I want us to be family – so comfortable with each other that we can laugh, cry, argue, fight, play, maybe not talk to each other for awhile, but then pick up right where we left off. That might sound silly, but there is truth in it. How do I keep the formal understanding, but ditch the formality?
    Thanks so much for sharing this!! Let’s keep exploring. I’ll keep you posted about what the kids in my neck of the woods say about shape.


  2. It’s a funny thing, definition, isn’t it. It might be interesting to ask for one for a non-mathematical category, just for comparison. What is a dog? A chair? Red?
    I think it’s very valid to ask, and a good learning opportunity for the students too, but something in it intrigues me… Perhaps that the definitions seem to reflect how the word has been used in a school context.
    Most of the definitions were about closed shapes, which is the way we usually talk about them in maths lessons, but outside school we use the word more broadly: ‘There was a weird-shaped mark on the door.” “What shape was her nose?”
    WODBs are such a good way in here, forcing the students to take a side, to make distinctions. So, if there’s an open shape, as well as 3 closed ones, students might name the open-ness as a feature, or they might say, “It’s not a shape.” And then there’s maybe a debate…


    1. It’s an interesting point. I think generally our definitions reflect the context that we encounter the words in, in a way our social landscape forms our linguistic one, our experiences and interactions shape (this word again) our not-so-platonic ideas. I wonder if with the development of socio-linguistic awareness students start selecting “appropriate” understanding for the context. They might even think about the words differently depending on if they encounter it in math class, on the playground or in their afternoon art school.
      I guess I’m leaning towards Wittgenstein’s “The meaning of the word is its use”. Every time we try to really define the word, we try to separate it from its context. In the beginning of the year, we got into a conversation about nature; students collected “nature” artifacts, sketched them, reflected on their own connection to nature. Then we came to the point of sharing the examples of what is nature, and then finally I asked them what does make all these things that they use as examples “nature”. “Something that is alive, not made by people, is everywhere, something we need to live, we are nature.” It was this “too broad to be defined” sense again.
      I like the focus on a particular attribute/attributes that WODB provides, thank you for reminding me about it. I will likely select a WODB prompt for my geometry conversation next week after we work on our hexagon controversy. I’ll see after tomorrow where the hexagon conversation takes us and which prompt would be more prompting.


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