Using End Pages to Inspire Math Conversations

"I think all of these have four wheels."

"No, the truck in the middle has 8. You just can't see the other four because they are on the other side."

"I like the red car best. It goes faster than the others."

"The red car has a number 5 on it. That's because it came in fifth place."

"No, the number five is because there are five cars on these pages. The other ones just don't have numbers on them."

 

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Have you ever taken time to appreciate the beautiful art that exists on the end pages of books? This year my goal has been to really slow down and enjoy books together with the children. Sometimes my literacy appetite is so great that I just can't wait to read all these wonderful books in my collection - we rush through book after book each day, gorging ourselves without slowing down to appreciate their intricacies. It's hard not to - there are so many amazing children's books in the world and I'm eager to share as many of them with my class as possible. However the educator in me knows it's important to take our time so we really get as much out of each book as possible. Appreciating the book's design, the illustrations, text, even choice of font, can all help empower children as well rounded readers. There is so much to be discovered within books when we appreciate each text.

Kassia Wedekind, co-author of Hands Down, Speak Out: Listening and Talking Across Literacy and Math, challenges us to think about "how we can mathematize end pages of books". She gives the example from the book Creepy Pair of Underwear by Aaron Reynolds. The end pages feature an array of mini underwear pictures (always a perfectly humorous topic for any elementary school student!). Kassia suggests that children can examine the array and use different strategies for calculating the total number of objects. Children can then be challenged to write different equations to show a deeper understanding of their thinking.

Looking to extend this idea we read Count on Me by Miguel Tanco. This book helps readers uncover the beautiful math that exists in the world around us (e.g., geometric shapes on playgrounds, sharing during dinner time). After reading the book for pleasure, I reintroduced it again to children and asked them to take notice of the cover and end pages specifically. They immediately noticed the vibrant pattern on the inside cover, and made the connection that a small portion of the pattern was also visible on the spine.

 

 

Participating in their discussion while honouring their ideas by including them in an idea web is always challenging for me. It's hard to listen attentively, talk, and record what is being said at the same time. I tried my best to capture some of their thinking on a large chart paper. Most children were eager to discuss the intricate crisscross pattern and curious about what other objects in our yard had this same design (e.g., many noted the fence also looked like this). Towards the end of our conversation the idea of quantity arose, and children wondered if the estimate of 30 diamonds was correct. Many felt this number was too small and thought counting by rows would be an easier way to find out. A next step for us might be to place the book along with different loose parts (e.g., gems, buttons), chart paper and a hundreds chart and challenge children to see how many diamonds they can count in the pattern. I also wonder if helping children to calculate a large quantity by creating groups of counters using a friendly number such as 5 or 10 might work. It's okay if we don't arrive at an answer to this question - the process of working towards it is just as important in my opinion. 

Interested in exploring end pages with children in order to provoke deeper math thinking and exploration? After you find a book that is meaningful for children and has beautiful end page art, consider using some of these question prompts: 

• What do you see?
• What do you think about ____?
• What does this remind you of?
• What do you like/dislike about these pages?
  
• Why did the author/illustrator choose this design for the inside of the book?
• What meaning does this end page have now that we've read the story? 
• What connections to this design can you make?
• What math do you see?
• What math questions do you have?
• How might we find an answer to your question?
• If you were the author/illustrator how would you have designed the end pages?
• How would you improve this design?
• How would you change this design to emphasize patterning/quantity/shape/colour/etc. more prominently? 
  
• Can you draw your own unique end page for a book you've written?
• What would you ask the author/illustrator personally about this piece?

Connecting math and literacy is a wonderful way to help children appreciate how connected the world is and find authentic problems to explore in their lives!

Risky Play Inspired Math

"This stump is kind of wobbly. The first two are strong but when I step on the last one it wiggles when I stand on it." 

"I'm a little nervous to step on it. I'm not sure if it will tip, but I want to try and see if I can do it."

 

"Watch me! This stump is so tall that it helps me go super high when I jump off."

 

"I'm the tallest in our class when I stand on the big stump. I'm even taller than Mrs. McLennan!" 

The children were playing outside. I was fascinated as I watched them bravely climb onto different sizes of stumps lined in a row. The first two were solid and level but the last was visibly uneven. As children stepped onto it, it teetered and moved under their weight. Because it was much taller than the other two stumps, it was difficult for children to step onto, making it a challenging physical feat. Children lined up and attempted to balance on the uneven stump, jumping higher and farther from it with each subsequent attempt. By the end of our outdoor play time there was a line up of children waiting patiently for their turn - each eager to see if they could outperform their last big jump. 

Risky play, like climbing and jumping off stumps, is incredibly important for children. They feel empowered as they engage in difficult tasks, building their physical and mental endurance. Children persevere through difficulties, building their grit and stamina and feeling untouchable in their moments of triumph. When children are successful in risky tasks, they take pride in their achievements and feel unstoppable. They feel inspired to tackle new opportunities and go further in their explorations. This helps build their confidence and sense of self. They feel like they can do anything!

As a big believer in risky play I recognize the many personal and social opportunities for learning that exist in challenging activities. However watching the children explore the stumps, I recognized that risky play also provides multiple opportunities for authentic and engaging math learning. Children are often eager to mathematize their achievements by calculating how fast, how far, or how high they have moved. As the children climbed and jumped, they were discussing balance, height, length, and time. They counted how many children were in line for a turn and discussed their place within it. They created a system for measuring the height and distance of each person's jump, and tracked these to compare growth. They strategized for how to improve their performance and planned for creating a more difficult obstacle course using additional stumps for future outdoor play times. A simple natural object like the stump created opportunities for rich STEM learning that were highly motivating and meaningful for the children. The math they explored was child-centred and authentic, and deeply connected to their immediate world. 

It's easy to understand why educators (and families) shy away from risky play. We worry about children getting hurt and the consequences of these actions, especially in a playground or school setting. But depriving children of risky play sends the message to children that they might not be capable of achieving a goal. They may misinterpret our fear of the experience as a mistrust of their abilities. These hidden messages may result in children wondering about their own interests, strengths and needs, and fearful of taking other risks in their lives at school and beyond.

Encouraging and supporting risky play sends the message to children that as adults we trust and support them in difficult tasks. We acknowledge that they might be nervous or apprehensive to try something new, but let them know we are here to help them through the experience. Children develop a growth mindset and embrace mistakes and missteps as learning opportunities when they overcome an obstacle. 

Co-creating shared understandings and expectations for children regarding the play is a great way to get started. Have an honest conversation with children about the benefits and risks of the activities, and brainstorm how to keep everyone safe throughout the explorations. As children gain experience in the activity they can add ideas to the overall expectations for play. Encouraging sharing afterwards can help children articulate how the risky play went - what worked well, what needs to be improved, and what can be done differently in the future.

 

Over the course of several days the children continued to play with the stumps. As I observed them I continued to notice math emerge in their work. Joining as a play partner helped me notice and name the math I saw, and scaffold and support their inquiries. I'm curious to see how their obstacle courses continue to evolve as we enjoy the lovely fall weather and spend more time together outside.

Mandalas

The children have been very interested in creating intricate, circular designs using a variety of loose parts and pattern blocks. We introduced the word 'mandala' to them. It has been interesting to see what shapes and patterns they can discover and identify. 

Texture Paintings

The children used texture plates today to create interesting designs in paint. They spooned paint onto paper and used the rubbing tools to mix and scrap.

Gorgeous!

Sand Textures

 Today we placed some fine sand from our sand table on the light table along with some texture plates. We invited the children to explore the materials and create interesting lines and designs in the sand.

It was so interesting to see the light shine through the textures as the sand became scraped away revealing the light table below.

Carving Prints

We love making prints with our children! Using styrofoam pieces and a sharp pencil, we encouraged the children to create their own print plates. 

The first step was to use the pencil to carve a design onto the styrofoam piece. We discussed with the children that the design had to be deep into the plate and encouraged them to trace their design with a finger to test and see if it felt deep enough.

After children were satisfied with their design, they rolled their plate with paint. We discussed how too little paint would not leave a dark enough print and too much paint would saturate the design.

The next step was to press the plate onto a piece of paper and carefully lift it, revealing the unique design!

The children created some amazing designs!

Tape Painting

Mrs. Bombardier put out a wonderfully layered painting activity for the children to explore. Children were invited to first cover a piece of painting paper with painter's tape (so it could be easily removed from the paper without ripping it). 

Once the children were happy with their tape designs, they painted on top of the paint using a variety of colours.

When finished painting, the children carefully lifted the tape from their painting, revealing an amazing design below!

Some children chose to repeat the process and put more tape on top of the painting, paint again, and remove the tape to reveal more designs. Others were happy with their original finished product.

The artwork lead to much rich discussion about the process of the art making and describing what the pieces reminded them of. The children had some interesting ideas. What do you see when you appreciate the art work?

One of our favourite blogs had a great post about how to talk to young children about art work that you might find interesting!