Eleanor Duckwork said, “Either they know it already and I’m too late, or I’m too early and they can’t learn it anyway.” (Reference 1) There are many routines in the math education world. Many offer great opportunities to support children’s development, however I want to reflect on this quote and think more deeply. Maybe stop here and think about a routine you use in your classroom (or if you are a parent, think about a math game or routine you do in the car with your kids). Reflect on why you use this routine and what might be (or not be) developing in your children.
When I began doing my wondering workshops in the fall, they were meant to open the doors of what mathematics is; what it feels like, sounds like and looks like. I wanted them to know and feel not just permission, but the freedom to share their observations and questions and know they were valued by me and their math community. I also wanted to support their language development in understanding the difference between noticing or observing and asking a question or wondering about something. They don’t always have this clear at the beginning of Kindergarten!
Recently, we embarked on some geometry work first using the Contexts for Learning unit Baby’s Wild Adventure. My students explored how a baby moved around exploring her backyard chasing after butterflies and toads. She kept dropping her blankie and finding it again. The kids drew maps and investigated what shapes parts of the backyard could be, and figured out how the baby kept finding her blankie again after moving down straight paths and turning corners. This allowed my students to think about shapes in ways that pushed them beyond thinking about the existence of just one triangle or one perfect rectangle. My students then designed their own playgrounds. We first looked at some aerial photos of playgrounds and then they set off to work using 3D objects to trace 2D faces.
After this unit, my students were given opportunities to play with different geometric building sets. Many creative endeavors ensued but one interesting thing in particular occurred. Some children were designing their own playgrounds with the building pieces. They used shapes to represent things like a sandbox, made pathways in certain shapes, or composed new shapes from individual shapes.
Many students began to notice the shapes made in the negative spaces.
They now request that I take “aerial photos” of their creations as they point out what things look like from above.
The children were quickly realizing they could compose shapes from individual shapes. For example, they could put two of the red trapezoids from the pattern blocks to make a hexagon. This realization occurred with different building sets. We then set out to discover our own new shapes. I wrote about this in my last blog post. This part was just plain fun. They loved “discovering” new shapes and naming them. During this work, they were beginning to count the sides and corners on their new shapes. This was interesting to watch as they would either tap sides or corners, or run their finger along sides. Of particular interest, they consistently counted sides and corners from a concave element.
Here is where I then decided to do a couple of wondering workshops utilizing shapes. I first invited my students to examine a couple of simple repeating patterns.
This was the first one they examined: (Reference 2)
Ivan said, “there’s always going to be more squares than hexagons.” I inquired, “how do you know that?” He added, “if you keep it going, you’ll always get more squares because you put two squares every time and only one hexagon.” He noticed this almost immediately and I thought it was an interesting numeracy connection. There is a 2:1 ratio occurring here. I had not even anticipated my students would make such an observation.
Next, they looked at this one:
Catharina said, “it looks like the green triangles are standing up and the tan diamonds are shadows.” Then Liliana added on saying, “yeah yeah, shadows, I wonder if you fold that up.” I asked if that was something we could investigate and she agreed. So I sent them off to replicate this repeating pattern with pattern blocks.
Right away, they figured out that folding the rhombus up did not match up to the triangle. Some knew that wouldn’t work without even physically having to do that.
Interesting, Catharina’s attempt to replicate this entailed the triangles standing up. I think she was seeing this in 3D.
Ivan was building the pattern out and very quickly said, “there’s the same amount of triangles and the same amount of rhombuses.” I noticed that he did not count either row and asked how he knew this. He responded, “because every time I put down a triangle, I put down a rhombus.” The big idea of one-to-one correspondence is emerging.
This made me have goosebumps! It was a lovely lead into our next Contexts for Learning unit (Beads and Shoes Making Twos) where the children began exploring different class sizes of children walking in two lines, and then different sizes of items that come in 2 rows like egg cartons.
Okay so lovely lead in was not planned ahead of time. It happened organically but it has given me many things to reflect on. How am I choosing what to put in front of my students and when? I want to be strategic and sequence tasks and activities with the goal of supporting what Cathy Fosnot calls “progressive development.” If I give them one activity, what comes next that will “up the ante?” As teachers, we can’t just do whatever comes next in our curriculum, or randomly choose a “neat” thing from a no-doubt excellent resource. There are many wonderful resources for selecting rich tasks. Something to explore further is how to sequence what we give children.
This is clearly not the end… stay tuned.
References:
1: Duckworth, E. 1987. The Having of Wonderful Ideas and Other Essays on Teaching and Learning. New York: Teachers College Press.
2: The repeating patterns were made using the Polypad on www.mathigon.com.
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