Invested Children = Robust Math Explorations

Research tells us that when children are empowered by adults who believe in them, a positive mathematical mindset, and robust math tasks (low floor, high ceiling) anything is possible. Reflecting back on the beginning of our week I certainly wouldn't have predicted the rich math learning opportunities and problem-solving that our children delivered. Here is the story of what happened...

I have been doing much professional reading about mathematical mindset (Jo Boaler, www.youcubed.org) and saw the following pattern in one of the books. I was fascinated by the math potential of the question and thought this activity might be something that would be of interest and a challenge for our children.

 

After presenting the question during our morning message, we tweeted our ideas about the pattern and figures. Two classrooms from elsewhere in the province decided to also explore the problem in their classrooms, and tweeted us back their ideas. Using social media to share our mathematical thinking and collaborate with other children grows our ideas and abilities as mathematicians as we learn there are many ways to communicate with others and that when we work collaboratively on a task, the outcome is greater and our understanding richer.

 

We discussed the ideas shared by Ms. Van der Veen and Ms. Ralph's classes and added their ideas to our own understandings of the pattern. This is what we noticed - the pattern grows and we can represent this with a mathematical equation (1, 1+2, 1+2+3, 1+2+3+4, etc.). We noticed that the base was growing by one each time and that the number assigned to the figure (e.g., figure 1) matched the number of blocks in the base and edge (e.g., figure 5 had 5 blocks in the base and 5 on the left edge). 

To help us test this theory we used blocks to build what we noticed. We extended the pattern and built figure 5 and 6.

I asked the children to think about the many other figures that might be created in this pattern. The children wondered what the 100th figure would look like. We decided to set this up as a math exploration for the children today. We provided photos of our previous work, the chart paper reference page, mini wooden blocks, and writing materials.

The children started by building the original figures in the problem. They realized that it was more effective to build the figure flat because building upwards wasn't reliable. The figures would easily tip.

One child immediately went to work on building the 100th figure. She understood the pattern (the 100th figure would have a base and edge each of 100) so she set to work.

This task required great patience and perseverance because the blocks are small and move easily. Putting them back in a straight line and counting and recounting to ensure that there were 100 took great commitment.

Other children became curious and stopped by. "Is the base longer than you?" he asked and we measured to find out.

Once the base was completed the left edge was started. Again, counting out 100 blocks and lining them up in a straight edge was a complicated task and it required great patience and persistence with the task.

She did it! Yeah! This is what the 100th figure's base and edge would look like! Now onto filling it in. Would we have enough blocks?

Another child thought that drawing the figure on a paper would be a better idea because he didn't think we had enough blocks to fill it in. He began, carefully counting out 100 squares to represent the base...

...and learned his paper wasn't big enough. He problem-solved by taping the papers together...

 

...and together and together and together...  Mrs. Tullio stopped by to assist. She was interested in learning about what the children were doing and they were so eager to share their thinking with her. They recognized the significance of this project. 

Once the base and edge were counted out and drawn, we decided to draw the other outside edge. Because we used chart paper and lined the papers up when taping them together, we didn't feel we needed to draw each individual block inside the figure.

Other friends stopped by to watch the work unfolding and to double and triple check to ensure we had 100 squares in our base and 100 in our edge.

 

He did it! This is what the 100th figure would look like! You can tell just how large this is with him sitting inside. Each paper is an easel sized paper and the total papers taped together is almost the size of our large carpet. Taking our time to think slowly and deeply about this problem, and working together with people both inside and outside our school, helped us to accomplish an incredibly impressive feat for kindergarten children. We couldn't be more prouder and wonder what problem we will solve next!

Exploring Insects

We have noticed many insects in our play yard. The children spent time finding them and gently catching them so we could explore them up close. 

We placed them in separate Petri dishes...

...and placed them on our light table for observation up close. 

Because the insects were contained reluctant children could safely explore without fear of the insects escaping or touching them. 

The light also illuminated the insects so it was easier to see their body parts. 

Virtual Classroom Tour

We have been so excited by our recent classroom changes. Here is a virtual photo tour for those of you who haven't visited in a while. 

Our art area has many interesting materials for the children to use. We believe that there is beauty in the materials so the way we present our art artifacts also helps to decorate the centre. Children's work is on display in frames and documentation of recent experiences is presented in annotated photo collages and learning stories on a clipboard, accessible to both classroom visitors and the children.

Our snack centre is a free flow, open space where children can visit during a playblock when they are hungry and would like to socialize over a snack. There is always a survey or writing component on the easel for children to read and respond to, and needed materials (napkins, spoons, etc.) are housed on a shelf nearby to encourage independence and responsibility.

Our science area has accessible tools (magnifying glasses, tweezers, etc.) and artifacts (shells, feathers, etc.) nearby for the children to explore. We love our light table and use it regularly and try and incorporate our current inquiry (in this photo the planting along the window) into the centre. We weave literacy and numeracy into all centres and there is much student writing on display.

Literacy has a wealth of spaces and materials to support our children's emergent reading, writing and oral language interests and abilities. The co-created alphabet was made from paper mache and our children use math/sorting skills when keeping the writing materials neat and organized. A bin of journals is available anytime a child wants to do some drawing and writing.

There are cozy corners in the room to sit and chat with a friend, read a book, draw in a journal, or play with a quiet bin.

Our math area has open-ended tools available for both focused table top activities and also integration into other experiences (e.g., measuring tapes to see how tall our plants have grown at science). Because we have focused so deeply on math we have woven much documentation of student work as well as the pedagogical and philosophical frameworks behind our activities into the display. The documentation offered at every centre is available for both children's and visitor's use.

Our drama centre changes based on children's interest and is currently set up as a mini house. The children help us co-construct the environment by gathering materials from our room and creating what we don't have.

Our cubby area celebrates each child with a recent framed photo and a self-portrait painted by each child.

Complex Patterning Problems in Kindergarten

We have done a lot of work this year in conjunction with our school improvement plan regarding helping improve children's attitudes and achievement in math. Part of encouraging a mathematical mindset has involved communicating the message to children that we believe in them and know they are capable of amazing mathematical work.

Recently we have introduced some interesting math problems to children and it's really encouraged some deep reflection on my part as an educator, especially when considering the types of questions I offer to children and how I support and scaffold authentic problem-solving and mathematical work in our classroom.

Take this problem for example. Previously I would have provided this type of patterning prompt during whole group time (probably in our morning message) to children and encouraged them to identify and continue the pattern in each example. I might have then provided the round stickers to children and invited them to create their own simple and complex patterns. When I think about this type of activity, it's satisfying what I was hoping to do (which is explore and create patterns) but I am not really sure that it's as challenging and open-ended as it could have been. 

After reading Jo Boaler's work on mathematical mindset I've been revisiting the types of problems and math situations we introduce to children. Yesterday I asked the children to consider what they notice about the following figures. 

We spent the first day discussing what we noticed and today we really delved deeply into the problem. We discussed with children what a robust math task is, and how we can spend a lot of time slowly and deeply thinking about a problem. We don't have to think of the answer right away, and we might all see and think something different about the problem. If you examine the children's responses today, you'll notice that they include patterning, measurement, number concepts, and addition in their observations of the growing figures. Some children also considered what was happening if they looked at the figure to the right first, and went right to left. They observed that meant the size and pattern was decreasing instead of increasing. Our conversation was so interesting and we are inspired to revisit this again tomorrow, hopefully using real blocks to actually build a model of this problem. By observing, discussing, writing and representing the problem we are differentiating the math and providing multiple entry points for children into the problem. We are on the same path towards deep math understanding and it's an incredibly exciting place to be!

We like to share what we do in our classroom using social media and often tweet about our learning. It was very exciting to get a response from another kindergarten class who is planning on exploring the same problem we did. We are hoping to tweet back and forth with this class so we can share what we observed about the math figures and learn what they notice. We can work together to collaboratively understand the math situation on a deeper level and use technology to help us communicate! Math is about collaboration, not competition!

Science Area

Although in our room we encourage children to use ideas and materials in cross-curricular ways, we have a designated science area. We display and store materials, tools, documentation and inquiries in progress here. We just cleaned and restocked the area so it is fresh for the remainder of the school year. 

Natural artifacts of wonder that are found in our yard or donated by families sit on top of shelving holding building toys. 

A mirror with a gardening trellis displays interesting photos. 

Extra baskets are always handy and incorporated as part of the decor. 

The current interest or inquiry is interwoven with children's art and writing. Here our recent planting is displayed in the window. 

A low table and comfortable pillows invite children to explore with various loose parts. 

Documentation explaining our philosophy for science and outdoor play are on display. Users can scan the QR code and be brought to a link containing all our blog entries tagged 'science' to see the history of the center and the stories of what the children have done so far this year.