On Wednesday, I saw a tweet from **Santina Fantetti**: a Kindergarten teacher with the Toronto Catholic District School Board. At the time, the tweet seemed like a straightforward question to me:* is a piece of paper 2-D or 3-D*? I was staring at some photocopy paper at the time, and it certainly seemed flat to me, so I was sure that it was two-dimensional. Little did I know that I had a lot of learning to do. Between the time that Santina sent the tweet on Wednesday until late in the day yesterday, the Twitter conversation kept going (*as you can see in the Storify Story below*).

These past couple of days, I’ve realized what this discussion really means.

- If you can’t hold something that’s 2-D, then what does this mean in terms of manipulatives?
- I like to think of math in terms of real world applications (thanks to one of my previous vice principals,
**Kristi Keery-Bishop**). What are the real world applications then for shapes? I’m thinking of Visual Arts, blueprints, and maps, but is there anything else? - For many reasons, I struggle with
**the value of worksheets**. Seeing the connection between “shapes” and “paper,” I wonder if a worksheet will become even more tempting as the best way to teach this concept. How can we move beyond the worksheet and make this learning more concrete for students that need it? - I look now at the
**Ontario Curriculum expectations**for Geometry and Spatial Sense. In numerous grades, the specific expectations discuss using “concrete materials” to sort and classify shapes. What would these concrete materials be, if shapes technically can’t be held? Is our curriculum in line with the mathematical definition? **Anamaria Ralph**‘s comment about**3-D solids and exploring faces**, made me wonder if we really should be teaching 2-D shapes in the context of 3-D solids. Then we can use manipulatives while also exploring the concept of 2-D. What do you think?

Days later, I continue to think that I owe many students an apology. In my fifteenth year of teaching, I’m gaining a new understanding of 2-D versus 3-D. For years, I’ve told students that two-dimensional shapes are “flat?” Now I’ve found out that I’m wrong. **But what**** are they then? How would we define them for our youngest to our oldest learners? **This multi-day discussion proves to me that we really are life-long learners. **What have you learned this week? How will this learning impact on your practices? **Let’s celebrate some new learning together!

Aviva

These questions, and perhaps more importantly, the willingness to ask them and wrestle with them, are doing the job. Thanks for sharing this episode of questioning and rethinking what you know. It’s a great example for all of us.

Thanks Sherri! I’m hoping that the questions eventually lead to some answers. At the very least though, they are leading to some rethinking of practices, and I think this will be beneficial for our students.

Aviva

I really enjoyed the evolution of thought you have shown in the Twitter conversation and your reflections here after. I think it will draw more conversation and ideas and we can all look forward to seeing part two of this post when you figure it out. I think you are on the right track by trying to define what 2D/3D is by the applications. Using those examples and working from there will help you clarify the conditions of each. I find that a lot in math: the examples help to explain the conditions and limitations.

Can’t wait to see where you go from here with it!

Thanks for the comment and the vote of confidence, Kristi! These 2-D and 3-D connections make the most sense to me, and I can’t help but wonder if by linking both, students will actually get a clearer and deeper understanding of each. We have a few provocations for early next week that involve 3-D figures, so maybe we can start the conversation in class from there. I’m very appreciative of educators on Twitter that have also shared some ideas for me to consider. This is definitely still a work in progress, but knowing my propensity to blog, I’m sure another post will come soon enough. 🙂

Aviva