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!

10 Math Concepts that Children Learn from Puddle Play

"Childhood is that state which ends the moment a puddle is first viewed as an obstacle instead of an opportunity."

K. Williams

 

Puddle play is an incredibly rich learning activity that many children enjoy. Spending time outdoors exploring many natural elements helps children recognize that there is no such thing as bad weather. There is beauty and wonder to be found in every aspect of nature. Sometimes in early childhood education we need to help our families and school community understand the authentic learning that can happen when children play with sensory materials like puddle water outdoors. Demonstrating the meaningful math connections that emerge in this type of play can build support for outdoor exploration and learning. Sharing this through regular communication including documentation can be incredibly helpful in cultivating positive partnerships and support for messy play outdoors.

Here are ten math ideas that can be introduced and strengthened when children play in the puddles:

1. Patterning

When rain drops fall into water their impact disturbs the surface tension of the water. The ripples spread outward from this impact point. This reaction forms concentric circles which are two or more circles that have the same center point. Each subsequent circle is larger than the last, creating a growing geometric pattern for children to explore.

2. Opposites (Float and Sink)

Children love to place objects in water and experiment with what happens to them. A favourite outdoor activity for our children is to place many different objects in water and see if they float or sink. An object's density determines whether it stays above or below the water. The object will float in the puddle if it is less dense than the water. If it sinks, it is more dense than the water. 

 3. Temperature

The temperature of puddles can vary depending on the ground and air temperature. Children enjoy feeling the water with their hands and describing how hot or cold it seems. An engaging activity is to provide kid-friendly thermometers to children and challenge them to read the temperature of the different puddles in the yard. Are the larger ones a different temperature than the smaller ones? Does the temperature of a single puddle change over the course of the day? Ask children to generate theories about why this is happening.

4. Measurement

Sensory experiences are vital for children's growth and development. Most children love to play in mud using cooking items and utensils. A favourite activity for us is to use the puddles for water play. Not only do we not have to worry about a wet floor indoors, children are captivated using cups, spoons, funnels and bowls to collect the water. Challenge children to measure how much water they can collect. Ask them to see if they can empty a puddle and calculate how much water it had altogether. 

 

5. Cause and Effect

Cause and effect activities help children realize that every action they take has a reaction. When children jump in puddles, the water splashes. Experimenting with cause and effect helps children play with variables that can control the reactions in different ways. For example, and bigger jump in the puddle usually results in a larger splash of water. Children can change their actions (e.g., increasing movement, decreasing movement, modifying direction) and observe the results.

6. Comparison

Puddle water can look many different ways. Some puddles are clear while others are muddy and filled with debris. Children can travel the yard and observe/describe what they see as they compare puddles to one another. They can also experiment with the different materials and observe the reactions that occur. For example when water is added to soil, the soil appears a darker colour. This occurs because wetter soil has less oxygen compares to drier soil. Some puddles are so saturated with dirt that there is a layer of mud that settles on the bottom and a layer of water that has risen to the top. As children explore different puddles they can notice and name what they see, and compare the properties of each to one another.

7. STEAM (science, technology, art, engineering, art, math)

STEAM challenges are highly motivating for children. Puddles offer many opportunities for educators to ask children to design and build an object to be used with the puddles. A favourite in our classroom is to ask children to create a puddle boat from loose parts (e.g., wooden craft sticks, cardboard, foam, aluminum foil) that floats. Another highly motivating construction activity is for children to build a bridge over the puddle for the mini cars to use to cross the puddle. 

8. Counting

In our yard we often have very large puddles when the rain falls due to the slope and drainage of the playground pavement. A fun activity is for children to crowd in and count how many can fit in the area of the puddle. Great math questions emerge in this activity - can the same number of children fit in the different sized puddles in the yard? How many boots altogether are in the puddle? Can we count the boots by 2s?

9. Reflection

 

Water is a reflective surface. When the water in a puddle is still, the surface is flat and can easily reflect light. If the wind is blowing and ripples appear, the reflection can become distorted. The harder the wind is blowing the more distorted the reflection appears. This type of reflection also offers an exploration of symmetry as children can identify the line of symmetry and see how each side appears. This symmetry is often curious for children who notice that the reflected side appears lighter/faded than the real object. Children can experiment with reflective symmetry by placing and moving different materials into the puddle and seeing the result.

10. Area and Perimeter

The size and shape of a puddle can often inspire conversation about its area and perimeter as children wonder about how big or small it might be. Sometimes we will use loose parts to help us measure (e.g., How many rocks fit around the outside of the puddle?  How many leaves can float on top of the puddle?). The distance across a puddle is also interesting to measure as children leap across the puddle and measure how far they've jumped, or build bridges to help the mini cars cross the puddle.

 

What other math have you explored in the puddles? Tweet me and share your awesome ideas and experiences! @McLennan1977

Looking for a resource to read with kids to support authentic math play in the puddles? Check out my new book Puddle Math: See, Think and Wonder! 

 

     

 

Most Requested Presentations

Hello everyone! I have been getting many requests for presentations lately. I love working directly with educators, especially in live workshops. There is something magical about helping people (re)discover a love of math. Due to Covid restrictions and my busy schedule, presentations can sometimes be difficult to plan. So as requested, I'm sharing my three most popular presentations here for those who are interested. I'd love to chat about what you are doing in your districts, schools and classrooms! Send me a message or tweet/instagram me @McLennan1977 and we can connect!

 

Books to continue the learning:

 

Embracing Math: Cultivating a Mindset for Exploring & Learning

 

Kindercoding Unplugged: Screen-Free Activities for Beginners

 

Joyful Math: Invitations to Play and Explore in the Early Childhood Classroom

Uncovering Math Using Observational Drawing

 "When we as teachers are purposeful about bringing nature into our classrooms, we create opportunities for curiosity, questioning, observation, collecting and analyzing information."

(Nenia & Clucas Walter, pg. 8)

I recently read the article 'Exploring Nature Indoors with Observational Drawing and Scientific Thinking' by Kendra Nenia and Melissa Clucas Walter in the Fall 2021issue of Teaching Young Children. In it the authors discuss the many literacy benefits for children who carefully observe and draw objects from nature. As I was reading I was also considering the rich mathematical wonderings and connections that observational drawing can offer children. 

Observational drawing is when children draw what they see. If they are observing and exploring a natural object they are creating a realistic portrayal of the object in a drawn format. Children can sketch black and white representations using only pencil or marker, or draw their outlines first and then add colourful details after. 

       

As children explore ask them to consider what they see. Use varied questions to prompt a deeper exploration of the object including:

• What shape is this object?
• What size is this object?
• What textures do they feel when they handle the object?
• How heavy/light is the object?
  
• Are there any patterns on the object?
• What colours can be seen on the object?
• Does sunlight or artificial light affect the object? 
• Are there any shadows created on the object when it's placed in the light?
• What contours can be observed? 
• Of what does this object remind you?
  

Ask children to consider how they can draw the object in order to highlight their observations and ensure they are represented. What other drawing tools can be used to help in this artistic representation (e.g., pastels, watercolours)? What math is uncovered in this process? Engaging in supportive discussions with children throughout the drawing can help them to 'notice and name' the math they see. Educators can enhance this mathematical understanding by scaffolding the conversation and highlighting additional math concepts (e.g., proportional reasoning, perspective, reflections, symmetry) and encouraging children to include these in their drawings.

Once a child is satisfied with his or her drawing, encourage the child to represent the object in another way using a different artistic medium such as clay, Plasticine or wire.

When we tried observational drawing we first examined and drew a small pumpkin.

               

We also explored dried hydrangeas from the garden.

      

As children draw, ask them to look back and forth between their drawing and the object to ensure they are including any interesting nuances that will help give character to their artwork. Remind children that there is no right or wrong way to create art, and that repeated practice over time (even with the same object) helps us to refine our skills. Observational drawing should not be rushed. Encourage children who might become fatigued or frustrated to take a break from their work - they can tuck their object and drawing away and return to it when they feel inspired again.  

 

The interest in observational drawing in our classroom continued as the children became curious about some different leaves that had been collected and donated to our science table. Notice how the children became comfortable drawing the object using pencil first, and then added additional colourful details to show the shapes, lines, and patterns they saw.

Once children are satisfied with their drawings encourage them to share these with their peers. Children can reflect upon their experience - How did their mistakes help them to learn and improve their work? What did they enjoy about the experience? What might they do differently next time? How might children like to display their work? Sometimes children enjoy gifting their art to others, or displaying it in a public space like the school library. These drawings can be added near the shelf where books about nature are housed. Perhaps children will be inspired to create a documentation display to share the process of their observational drawing, and the final pieces, with others in the school or community. 

 

Observational drawing is an incredibly rich artistic experience that is infused with science, literacy and math. As each season brings new treasures, it can be revisited over and over throughout the school year.

Autumn Math Walk

 “I'm so glad I live in a world where there are Octobers.”
― L. M. Montgomery, Anne of Green Gables 

What a beautiful time of year it is in Southern Ontario! We have been spending much time outside each school day exploring the changes in our school yard. The children are curious about the world around them and ask many questions about their surroundings and the autumn treasures they find. Many of their wonderings have been mathematical in nature - Why is there a pattern on the sunflower's seed head? What are those holes in the stump and where did it come from? Why is a Daddy Long Legs an arachnid and not a spider?  How do leaves change colours? Why are the clouds moving so fast in the sky?

There is math everywhere. If one looks deeply enough, there is always a mathematical connection in nature. The sunflower seeds follow the Fibonacci Sequence - a numerical pattern meant to help maximize space and fit as many seeds into the head as possible. The perfect little circles in the stump might have been caused by a wood boring insect or bird looking deep into the bark for food. Daddy Long Legs have a different body than spiders even though they have eight legs. Leaves change colour because the amount of sunlight they need to create their own food lessens in cold weather (as the Earth's axis tilts us away from the sun), resulting in a chemical change in the leaves. 

I've been motivated to continue to look for rich mathematical wonderings and opportunities for inquiry each time I'm outside - even on the weekend with my own children. I find it fascinating to see math applied in an authentic, interesting, real world connection. The math in nature is beautiful! Inspired by this mathematical beauty I have written a book called Autumn Math Walk that includes photos and narrative to help readers delve deeply into exploring math in the outside world. I've appreciated the interest in this book and loved seeing the ways educators have used the text in their classrooms. Learning from one another is so inspiring!

 

    

I'm so very excited to partner with Pembroke Publishers and offer a free teaching resource to complement my book Autumn Math Walk!

Autumn Math Walk Teacher's Guide is a free resource that can help educators understand the math that exists in the natural world, and use this information to provoke conversation and understand with children as they explore nature. Perhaps children will be inspired to author and illustrate their own Autumn Math Walk book unique to their specific surroundings!

Pumpkin Math

Autumn is such a wonderful time of year! The colours and textures can inspire such rich math conversation! This past weekend we visited a few farm stands in the area and all I saw was math around me! I'm hoping that these photos will be useful as math prompts for 'math talk' conversations with children.

Show children a photo and ask them to share what they 'see, think and wonder' about each photo. The 'see, think, wonder' thinking routine can be used to help children delve more deeply into exploring and articulating what they notice in a specific situation. As children notice and name what they see, they can make connections to their own experiences and ideas, and form questions for further exploration and math research. 

Educators can also ask children specific math questions to guide their thinking. This can be especially helpful if the math concepts in the photos are geared towards an existing exploration the children are having, or if specific curriculum and assessment investigations are needed by the children. Sample math questions are provided below!

How many bumps are on this pumpkin?

 How big around is the stem?

What patterns do you notice?

How are these gourds organized?

What other ways can you sort the objects?

How many do you think are in each bushel? How do you know? 

 

What patterns do you see?

Why is the pumpkin more narrow at the top?

How big around might this be in the largest part?  

How many pumpkins do you estimate are in this box?

How many pumpkins might fit in the box if it was filled to the top?

How much money do you think the farmer should charge for the entire box of pumpkins? Why?

If these pumpkins are $3.50 each, how much would all of these cost? 

Should bigger pumpkins cost more than smaller ones? Why?

Which do you think is the biggest? How do you know?

 

Which pumpkin would you pick if you only had $2? Why? 

If you had X dollars, how many pumpkins could you buy?

How much money do all these pumpkins cost altogether?

What symmetry do you notice?

What patterns do you see?

How many pumpkins would a full crate hold?
 

Which gourd does not belong? Why?

How could you sort these? What's your sorting rule?

How many gourds do you think are in this bushel?

 

Halloween Math uses the 'see, think, wonder' thinking routine to explore seasonal objects and experiences in a mathematical context and is now available from Amazon!

Joyful Math Trailer!

In Deanna Pecaski McLennan’s kindergarten classroom, math isn’t limited to a specific block of time. It’s built into the environment and inseparable from everything her young students do. All of the math is infused with a sense of exploration, wonder, and joy.

Deanna’s book, Joyful Math, is about creating invitations for young children to engage with math ideas through art, literacy, and outdoor play. She focuses on building spaces in early childhood classrooms where children see themselves as mathematical thinkers with valuable ideas from the very start.

Joyful Math is filled with a range of tools and models, including:

• stories, vignettes, and photos illustrating how to develop a classroom environment that fosters curiosity and wonder for mathematics
• practical tips for inviting students to engage in mathematical play throughout the day
• examples of ways to document children’s experiences to make math learning visible to parents and the greater community

Supported by her experiences exploring math with young children, Deanna’s methods will inspire educators to be curious about math, take risks, try different approaches, observe carefully, and collaborate with children as co-learners.

Get a copy here!

Stenhouse Publishers (United States)

Barnes & Noble (United States)

Pembroke Publishers (Canada)

Amazon (United States and Canada)

15 Easy Autumn Invitations for Learning

Autumn is a beautiful time of year! There is so much potential for encouraging rich, playful exploration with natural objects that children find fascinating to explore. In this post I will share 15 easy to create autumn invitations for learning that can be presented in the classroom as table top activities, or investigated during outdoor play. These sparks for learning integrate math, literacy and science while offering potential for additional inquiries depending on how children explore and manipulate the materials. 

Educators can observe each activity and provoke further discovery using a 'see, think, and wonder' learning routine. After watching the children play with the materials, ask them to share what they are noticing while playing by stating "I see...". They can then be asked to reflect upon their observations and begin to build their theories about the play by making connections in an "I think..." statement. Finally a question for further exploration can be crafted in an "I wonder..." open-ended query. As children pose questions and share their curiosities about the materials and the world around them, the educator can record observations, reflect upon how to support children in additional work, and gather materials to enhance and evolve the play. 

For example - "I see many different kinds of leaves. I think they might be different sizes. I wonder if I can order them from smallest to biggest."

Natural Objects Colour Matching - recycle paint chip strips by providing them to children and encouraging them to find objects in the yard or placed on a table that match each shade as closely as possible. Encourage children to explore the different shades of each colour.

Roll and Fill the Web - create a spider web by wrapping a shallow basket with thick yarn. Encourage children to roll a die (or multiple dice), add or subitize the pips, and add that many spiders to their web.

Spider Game - create mini catapults by securing a wooden clothespin to a solid block or piece of wood with an elastic. Children can manipulate the catapult and fire a plastic spider into the web. Every spider that lands on the web earns them a point! Children can record their points using tallies, and compare their scores to others.

Measuring Circumference - present a variety of interesting pumpkins and gourds to children. Ask them to suggest ways to measure the circumference of each object. Order the objects from smallest to largest circumference.

Counting Seeds - encourage children to use fine motor skills to remove the inside contents of a squash, gourd, or pumpkin. Provide a variety of tools (e.g., hundreds charts, ten frames) for the to use when calculating the total number of seeds they have removed.

Gourd Runs - present a variety of gourds and recycled floor boards to children. Encourage them to create a variety of ramps and runs to experiment with rolling the gourds down.

Geo-Pumpkins - encourage children to create their own shapes and designs on pumpkins by using pushpins and elastics. Children can push the pins into different places on the pumpkins and then stretch an elastic around them.

Gourd Catapults - present a variety of recycled tubes, wooden boards and mini pumpkins to children. Model how the tubes and boards can be used to make a catapult. Children can be encouraged to stomp on the catapult and launch the pumpkin as far as they can. Brainstorm with children how to record how far the pumpkin travels. Discussing safety rules beforehand is recommended (e.g., ensuring no one is standing in front of the catapult before launching a pumpkin).

Creepy Crawly Pick Up - provide a variety of seasonal trinkets to children in trays. Encourage children to pick up and move the materials using a variety of fine motor tools (e.g., chopsticks, tweezers). Children can sort the materials in the trays in different ways (e.g., by colour, size).

Roll and Fill - provide a template for children to use (grid paper also works). As children roll the dice and add the pips, they then colour the corresponding number of objects or squares on their paper. First to fill a line, section or the entire page wins. 

Big Body Spider Webs - encourage children to wind yarn or twine around natural materials outdoors (e.g., stumps, tree trunks, branches), creating a large spider web. If the web is large enough, children can move their bodies in and out of the web, avoiding the yarn as an obstacle course. Large plastic spiders can also be added and the web transformed into a dramatic playscape.

Pumpkin Weigh In - provide a variety of measuring tools (e.g., balance scales, digital scales) for children along with a collection of pumpkins and gourds. Encourage children to measure and record the weight of the objects. Provide challenges like asking children to find the lightest/heaviest pumpkin or ordering them from lightest to heaviest.

Hanging Leaves - attach yarn or twine between to objects (e.g., tree trunks, posts). Provide a variety of real or fabric leaves and clothespins. Encourage children to attach the leaves to the twine in a variety of ways (e.g., sorted by shade, in a pattern, smallest to largest). Children can explore the play yard for other interesting objects to hang on the line.

Natural Materials Sensory Table - fill a sensory table with autumn items found in the play yard or provided by families. Add tools including magnifying glasses, tweezers, scissors and sorting trays to enhance the play.

Which One Doesn't Belong - present a variety of different pumpkins to children. Ask them to explain which one they feel doesn't belong and articulate their thinking using math ideas and terminology.