Sunday, March 15, 2020

Spring Math

The last week has been interesting for those of us adjusting to a new kind of normal here in North America. Like many of you, my family and I had a whirlwind of a week - cancellation of a trip two years in the planning, notice of school closure for weeks after our March Break, no time visiting family and friends, and limits to how often we are travelling in our local community.

The first few days were surreal; I had a difficult time saying goodbye to my students on Friday, wondering how long it would be until I saw them again. As my teaching partner and I prepped our centers and morning message, we worried about when the next time we would see each other might be. I walked out of school bringing home my plants, teaching materials, and personal belongings, not knowing when I would return.

In times of uncertainty, helping one another is one of the best ways to get through the stress and worry of what awaits. I know that many educators and families right now are wondering how to help support children even when we can't be together physically.

I love math. I also love to help educators, families and children love math. I have decided to create this blog post in order to continually provide ideas for how children can explore math in their natural world. The CDC is asking us to engage in social distancing and being aware of what is recommended is important. Right now being outdoors in our yards, on trails, and in gardens is still safe and encouraged. I realize that some of us are limited by our personal circumstances and not everyone has access to a yard or natural trail. I will try my best to vary activities in order to meet as many circumstances as possible. I will also tweet ideas for math learning on a regular basis @McLennan1977.

My hope is that the website will give ideas to educators who might be providing virtual support to the children and families in their care (e.g., if you are providing home schooling ideas, if you are engaging children in virtual classrooms), or ideas for families who are trying to engage their children in a sense of normalcy by playing and exploring math together in the outdoors. Check here regularly for new ideas as I will add more every few days, and feel free to email me questions or suggestions

Here are some fun ways to get outside and see/think/wonder about the math you discover:

1.  Natural and Man Made Patterns
Go on a hunt and see what patterns you can see or hear (e.g., bird calls, veins on leaves, fence posts, brick designs, tile mosaics). Bring a clipboard and crayons and ask children to copy and extend the patterns they see. Ask children to classify the patterns as natural or man made.

2.  Same and Different
Choose two or more random objects from the outdoors (e.g., a rock and stick, a leaf and bird). Ask children to describe the characteristics of each and see how many similarities and differences they can find between the two objects (e.g., a rock and stick are both hard and nonliving; a stick is longer than a rock).

3.  Number Hunt
Choose a range of numbers appropriate for the age of the child (e.g., 0 to 10, 0 to 20, 0 - 100). Create an easy template for children to use that help them track the numbers they see. Go on a walk with your child and encourage him or her to record how many of each number they can spot by placing a check mark next to the corresponding number on their sheet. At the end of the walk calculate how many of each numbers were spotted and analyze your data (e.g., which number was spotted the most, least, the same).

4.  Estimate and Count
Before venturing out with a specific destination in mind (e.g., biking around the block, walking to the mailbox) ask your child how long it will take to get there (e.g., five minutes to bike one block, twenty steps to the mailbox). Decide how you will conduct an accurate measure (e.g., use a stopwatch to time your bike ride, count steps as you walk). Once you have reached your destination compare your estimate to the actual measurement. What do you notice?

5.  Shape Hunt
Take a sketch pad and pencil with you on your next walk. Encourage your child to spot shapes in the world around them. Ask you child to consider if these are natural or man made shapes (e.g., a round flower, a triangle in the climber). Encourage your child to sketch a picture of the shapes they find or take digital pictures that can be reviewed at a later date.

6. Shadow Play
Experiment with how your shadow reflects your exact movements by playing in the sun. Ask your child to consider whether or not the shadow is a reflection? Ask them to explain their thinking and justify their reasoning. Play with shadows at different times of the day. Digital photos can be taken and the lengths compared.

7. Wiggly Worms
Worms are a treat for many children to find on wet, spring days. Ask the child to closely examine how the worm moves, and hypothesize how it travels so easily without arms or legs. Worms move by constricting their body muscles in a pattern. Ask the child to closely examine the stretching and pulling motion that it makes while wriggling about.

8. Puddles
Children love to jump and splash in puddles. Puddles offer children many rich math opportunities too! Ask a child to determine how deep a puddle might be (e.g., looking at how high it is on their boot, using a stick). Ask the child to hypothesize how s/he might go about measuring just how much water is in the puddle. Watch the puddle over time and record how long it takes to evaporate. Note how much smaller it is each day by drawing the outline with sidewalk chalk each passing day and comparing the different rings to one another. 

9. I Spy
Encourage children to pay careful attention to their surroundings. While walking outdoors look for interesting designs. Play a math "I Spy" game with children using only math clues (e.g., "I spy something with a circular pattern." "I spy something that has a growing pattern.").

10. Bubbles
Blowing bubbles provides an opportunity to engage your child in rich discussions about open ended math questions: What is the biggest/smallest bubble you can blow? Why is the size of the bubble different depending on the force of your breath? How high/far do you think your bubble might travel? 

11. Unusual Numbers
Encourage your child to look for numbers in unusual places (like the 185 on this hydrant). Ask your child to identify the number, and hypothesize the purpose for the number. See if you can find other similiar objects with numbers too (e.g., find other hydrants and keep track of what numbers are listed on each).

12. Foil Textures
Look for interesting textures outdoors (e.g., tree bark, bricks, paving stones). Give your child a small piece of foil and encourage him/her to smooth the foil on top of the object, carefully revealing the texture as the foil is pressed and molds against the object.

13. Number Challenge
Keep track of the different numbers you find. Challenge your child to find specific numbers (e.g., the highest number, the lowest number, their age, an even number, an odd number).

14. Sorting Rule
Find an interesting collection of objects (e.g., pile of rocks, sticks, flowers in a garden). Ask you child to list as many different ways that she/he could sort the objects as possible (e.g., sort by size, colour, shape, function). If possible ask your child to physically sort the objects.

15. Different Number of Groupings
Challenge your child to find groupings of objects. Can you spot 2s, 3s, 4s, & 5s? Here we spotted 3 groupings of 3 grasses for a total of 9 similar plants in the yard. Encourage multiplicative thinking when asking your child for the total number of objects (e.g., Three groups of three equals 9.).

16. Tape Resist Art
You (or your child) can create an interesting geometric design with tape on a flat, outdoor surface. Fill each section of the tape with sidewalk chalk or paint. When each section is filled, remove the tape to reveal a beautiful piece of mathematical art!

17. How Many Steps?
Find a challenging walking surface, like the wooden beam at the edge of a garden. Encourage your child to walk one step after the other along the length of the beam. Count each step. See how many steps your child can walk before she/he steps off the beam. Challenge your child to reach the highest number she or he can!

18. Leaf Patterns
Look for leaves on your walks - new leaves are emerging with the warmer weather and many older leaves are preserved in piles gathered in landscaping. Turn the leaf over and examine the vein pattern you discover. Use crayons and paper to create rubbings of all the different patterns you find.

19. Measuring Stick
Create a 'measuring stick' by finding a sturdy branch outside and wrapping it in equal segments using a heavy duty duct or masking tape. This is a nonstandard measuring tool so as long as the segments are equal, it will work. Encourage your child to explore the outdoor space and look for different things to measure (e.g., the depth of a puddle, the height of a plant).

20. Outdoor Easel
Create an easy 'easel' for painting or drawing still life observations outdoors by using a recycled plastic hanger to attach a paper to the fence. The hanger prevents the paper from blowing in the breeze, and can be used to hang the painting until it is dry. Encourage your child to paint or draw patterns that are visible in the outdoor space.  

21. Real Life Arrays
Encourage your child to look for real life arrays while exploring the neighbourhood, Help your child reframe the array as a multiplication sentence (e.g., "Three rows of three is the same as three multiplied by three.") and ask you child to calculate the product.

Spring Math Walk

To help inspire children and provide background information regarding the math that is accessible in the natural world around us, I am making the Kindle version of my newest book Spring Math Walk free on March 19. This offer is only valid on the site. I will add additional dates in the future. I'm hoping that this free resource will be of interest for families looking for additional math information for their children - helping them venture outdoors and becoming inspired by the amazing math nature has to offer!

Tuesday, November 26, 2019

Encouraging Computational Thinking In Kindergarten

Young children are capable of engaging in rich computational thinking activities in the classroom. In emergent programs, the questions and wonderings children have about the world around them can easily be embedded into unplugged coding activities. Activities that incorporate computational thinking are often misinterpreted as 'more screen time' for young children, when in fact, all of the activities I use in my classroom to encourage computational thinking are screen-free without the need for computers or tablets.

Computational thinking is a foundation of computer science, and intricately related to many of the big math ideas and practices found in early childhood curriculum. It involves many concepts and approaches.

There are many different elements of computational thinking including:
  • using logic to predict and analyze when engaging in problem-solving
  • making judgments when faced with problems in order to evaluate what is happening and consider how to move forward
  • creating and following algorithms (or steps of action) in order to move through the problem solving process
  • using patterns in his/her work to find and use similarities efficiently 
  • deconstructing larger problems or actions into smaller, more manageable parts
  • filtering out information that isn't directly related to the problem or an efficient way of solving it

The computational thinker:
  •  tinkers with loose parts and other open-ended materials
  • designs innovative projects and follows through on creating them in some capacity
  • debugs by finding errors in his/her work and correcting them in order to move a project forward
  • perseveres and continues with projects even when they are challenging
  • collaborates with others as part of a team for the betterment of a project 
'Hour of Code' is December 9 to the 15 this year. Hour of Code originally started as a project to inspire people everywhere to spend one hour coding, in order to show that anyone, anywhere can code. It has become a world wide celebration of computational thinking and coding, and many global classrooms participate. 

Looking to engage children with computational thinking in your classroom? Here are some ideas to get you started. 

1. Use a read aloud to introduce the concept to children. Computational thinking can seem abstract, especially out of context. The book 'Think Like a Coder' introduces computational thinking to everyday activities that children experience including cooking, building, and even exploring the outdoors.
2. Look for everyday activities that naturally engage children in inquiry, problem solving, and computational thinking. Building with wooden blocks is a great start! Help children articulate the steps they took to build their structure, and ask them how they problem solved when difficulties arose with their creation (e.g., how they fixed a tower that kept falling over).

3. Provide STEM challenges to children. Ask them to use specific materials to achieve a certain goal, and encourage them to articulate the steps and criteria they used in order to be successful. In this photo a child turns a geoboard into a marble run maze. 


4. Introduce the idea of coding using arrow coding cards and positional language. Children can easily explore pathways by creating them using different building materials (e.g., blocks, legos, magnetic tiles) and then exploring direction of movement.


 5. Transfer coding into algorithms by helping children create and follow specific sequences of actions. Here children create and follow an 'exercise code' where they must look at each line of code in the script, and follow the movements precisely.


Computational thinking is a versatile, foundational approach to learning that helps engage children in rich mathematical activities. As Hour of Code approaches be sure to check out my twitter and instagram @McLennan1977 for more ideas on how to bring computational thinking and coding to your early years and primary classroom. 

Let's connect! Have a great idea to share? Leave it below in the comment section or tweet me and use the hashtags #kindercoding #joyfulmath #HourofCode and #MTBoS!

Saturday, October 26, 2019

Autumn Math Walk

"Notice that autumn is more the season of the soul than of nature."
 Friedrich Nietzsche

What better time of year to head outside and enjoy the beauty of nature than in the fall!

We have been spending long amounts of time each day exploring the treasures in our yard. The children have been fascinated by how the world around us is changing - leaves are changing colours and falling from trees, Monarchs are gathering the last bit of nectar from the garden before heading South, and we are enjoying the last flowers in our gardens before they wilt and dry.

The children have had many questions about some of the interesting objects they have found in the yard - stumps, leaves, creatures and flowers.


Many of their wonderings have been mathematical in nature - Why is there a pattern on the sunflower's seedhead? 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?

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! Consider going on your own autumn math walk and seeing what amazing questions and connections emerge with on your journey!

I'm excited to publish my first children's book Autumn Math Walk (Kindle edition) on amazon now. Stay tuned for a print version!

Saturday, September 21, 2019

Math and Literacy Family Bags

Many families are eager to help their children practice literacy and numeracy work from school. Although traditional 'paper and pencil' homework is not developmentally appropriate in many circumstances, providing 'take home' games and activities in the form of literacy and math bags is welcomed and appreciated by many families. These interactive, hands-on, developmentally appropriate activities are easy to create and target many emergent skills and knowledge needed by young children. They are a good alternative when the school community expects work to be sent home from school. Take home bags help parents and guardians understand how literacy and numeracy can be nurtured and celebrated in the lives of young children.

You can access these, along with many free math printables, here:

As always feedback and suggestions for future resources is always appreciated!

Saturday, September 14, 2019

Exploring the Outdoors Mathematically with the Five Senses

Stop! Listen!”

We were outside lining up next to the school ready to head in and start our morning when Carleigh held up her hand indicating that she wanted her peers to halt their conversations and pay attention.

“Do you hear that?”

In the distance I heard what had caught Carleigh’s attention. It was the joyful ‘chick-a-dee-dee-dee’ of a Black Capped Chickadee, a friendly native bird that enjoyed foraging in the yard adjacent to our playground.

“It’s a pattern!” Emme observed. “I hear it too! I hear the dee-dee-dee part over and over.”

The chickadee continued its happy song and we stood as still as we could and listened.

“I hear it too!”

“Chick a dee, dee, dee! Chick a dee, dee, dee!” Carleigh sang-song and within a few minutes the rest of her peers joined in, repeating the cadence of the neighboring bird’s early morning song. 

The natural world is waiting to be discovered and often the math that children first uncover is through mindful observation with their senses - sight, touch, taste, smell, and hearing. Children who emotionally connect with and nurture their surroundings will grow to be environmentally conscious and mindful of their impact on the earth. We spend at least an hour outside each day - this includes guided, teacher provided invitations and activities for learning, and also free exploration by children in the yard. This year we are also hoping to start 'Forest Fridays' where we walk the local nature trails in the community around our school on a weekly basis. The outside holds amazing learning for children. Helping children notice and name their mathematical findings by incorporating the senses increases their understanding and connection with the world. There are also many wonderful children’s books about the human body that can be used to introduce each sense and help children focus on paying attention to a particular sense when outside. ImageImage One of our favorite math activities is going on nature walks mindfully focused on one sense at a time. Children walk together and use technology to capture their findings (when going on a ‘hearing walk’ the children listened for different sounds in nature and we used the microphone feature of our iPads to capture what they discovered). Sometimes the math may begin to blend and blur with science topics. This is an important connection for children as they see the relevance and authenticity of math in their everyday life. It is also another way for educators to justify the time spent outside; integrating subjects such as math and science helps to fulfill more standards while making learning more meaningful and relevant to children's lives. So many children (and adults!) consider math (and science) to be 'textbook subjects' - ideas that are taught in a classroom in isolation from the rest of the world - when tangible and concrete exploration of these concepts is developmentally appropriate and helps children see the relationship between school subjects and their own lives.
Some suggestions of starting places include:
Sight *Ask children to observe the world around them. Using your eyes/sense of sight, what do you notice? What do you wonder?” These open-ended prompts help children to think deeply about their surroundings and notice and name small details. It also empowers them to ask the math questions about the world around them instead of following the teacher's lead.
*What patterns can you see and identify in our space?
*What shapes do you notice in your surroundings?
*Can you locate different numbers or groupings of objects? A pair? A group of three, four, five, etc. *How many different shades of one color do you see?
*Can you group objects based on a sorting rule?
*What’s a reasonable estimate for something large - how many leaves are on the tree? How many ants might be in the garden?
*Can you search the yard and locate/match similar objects based on how they look?
*Play a game of “I Spy” using math language and encourage children to make predictions on what object you are referring to (“I spy with my little eye something larger than my hand…. Something taller than our fence…) 
Touch *How do different objects feel? Can you describe them? What do you notice? What's the same? Different? After children have shared their observations you can also highlight the size, shape and texture of the various objects.
*Can you describe the object in a mystery bag by how it feels? Can you guess what it is?
*Run your hand along a patterned object and describe what it feels like.
*Can you sort objects based on how they feel (hard, soft, squishy)? 
*Encourage children to create a collage using natural objects they find in the yard. Identifying the object’s characteristics and then classifying and sorting helps children build data management skills. Children can discuss the inspiration behind their choices or their placement on the collage.
*Bring sculpting materials outside including play dough or clay and encourage children to sculpt it using natural loose parts. What prints can they make in the dough using different objects including pinecones, leaves or sticks? Can they see or feel patterns pressed in the dough? How do they describe these patterns to others? Can they continue a pattern? 
*Give children touch challenges - who can find the heaviest item in the yard? How many children does it take to lift this object? Is it even possible to lift it? How do children problem solve when they find an object that cannot be moved? Can they find the lightest? Can children order a collection of rocks by weight (lightest to heaviest) or flowers by shades of the same color (lightest to darkest)? 
*Ask children to feel different objects and describe their temperature. Are items hot? Cold? Can their properties change if heat or cold are added?  
Taste *Eat snack or lunch together picnic style - as children eat ask them to classify their food. What kinds of food do they have? Can they describe what they are eating using mathematical terms? (“This jam sandwich is so sticky I need to chew it ten times before I swallow!" "My celery stick takes five crunches to eat." "I lined up my crackers in a pattern!").  
*Harvest seeds, fruit or veggies from a garden and ask the children to count how many of each item they have grown. Can children measure to see how long their fruit or veggies are? Provide samples of each food and have children vote on whether or not they enjoyed the taste. Children can write their own recipes using these harvested foods and experiment with preparing or cooking them in different ways. 
*Offer children foods grown from the local area. After tasting samples of each ask the children to classify the foods using a common characteristic (crunchy, sweet, tangy, mushy, sour). Photos can represent each food and be placed in the appropriate place on a sorting ring or on a chart. 
Smell *Ask children to identify and describe the different smells they notice in various areas of the classroom, yard or school. What connections to the smells can they make?
*Encourage children to go on a ‘smell hunt’ around the yard and find natural items with different kinds of smells (sweet, strong, unpleasant). Children can place the object on a chart placed in a central location. After the hunt children can calculate how many of each item were found.
*Play a game of mystery smell. Place an object in a bag and have children close their eyes and smell. They can use language to describe the smell and then indicate whether or not they liked it by voting on a class chart. 
*Children love to create their own ‘potions’ outside. Often when objects are crushed they release a stronger smell. Encourage children to create their own recipes or smell equations by gathering different objects outside and mixing them together. Children can also use large rocks to crush and mix the objects together. Provide clipboards and pencils so children can write down their ‘equations’ (10 rose petals plus a handful of grass equals a sweet and strong smell). 
Hearing *Ask children to mindfully sit on a blanket in the middle of the outdoor space and close their eyes and observe what they hear. Can they identify what the sounds might be? How many different types of sound can they count? Can they guess the direction and distance of each sound? What patterns can they identify (bird calls, insect chirps). 
*Ask children to use their bodies to make different sound patterns. They can clap, stomp their feet, tap their knees, and snap their fingers in different rhythms. Play follow the leader where one child plays a pattern and the others have to repeat it. Children can articulate the pattern rules they hear (one clap, two stomps, one clap, two stomps).
*Study different animal calls - many have a repeating pattern in their sound. Play recorded animal calls on the iPad and ask children to describe what they hear. Challenge children to sit quietly in an area and try to get a bird’s attention by echoing its call. Record the noises you hear in the outside world and play these during quiet mindful time later on in the classroom.
*Create a musical wall by hanging recycled materials on a fence for children to explore - pots, pans, muffin tins, bells, metal pieces - and encourage children to create their own songs. These can be recorded as musical notations and ‘played’ by others. Encourage children to read other children’s music and play it using the materials provided.
*Provide a portable music source (wireless speaker, iPad) and encourage children to listen to and identify the different rhythms they hear. Add colourful scarves and other fun props to encourage big body creative movement incorporating patterning. Encouraging children to internalize and make emotional connections to the world around them using the five senses cultivates strong eco-stewardship and math learning. In addition to guiding children through the teacher-initiated games and activities listed above, there are many more ways to cultivate a mathematically rich and responsive learning environment in any outdoor space.

Monday, September 2, 2019

Cultivating a Math Rich Learning Space


It's Labour Day! This time of year always fills me with hope. Although it's sometimes sad to be leaving the fun and comforts of summer - spending time with my children, sleeping in, enjoying the back yard with my pup - it's exciting to think of the possibilities of another school year. Like a blank canvas waiting to be painted, our classroom evolves and changes over time as the children and I learn and explore together. 

As we head back into the school year many educators are working thoughtfully to create environments that are rich with math possibility. In Reggio Emilia the environment exists as the 'third teacher' inspiring, supporting, and extending children's learning in rich and complex ways. I have been contacted by many educators asking for advice on how to best set up their classroom and routines in order to create as many opportunities for authentic math as possible. This has inspired today's blog post - how to cultivate a math rich learning space for children at the beginning of another school year.

I thought it would be easiest to create this as a 'math check list' with points to read and consider. When I reflect upon my own math pedagogy and practice, these are what I think stand out as mathematically meaningful for the educators, children and families that share our space. Although this check list isn't all inclusive, I thought it might spark some ideas to support and inspire as we enjoy the final weekend of summer and start back in our classroom with open eyes, minds and hearts this week.

1. Make math a part of every space in the classroom and child's school world. In the classroom are there math tools and materials available for use beyond a 'math center or math shelf'? Do children see how math relates to every subject in the space (e.g., how materials are sorted and stored on the toy shelf, how measurement is used when children decide on a size of paper to use for their project)? Can they translate math tools and ideas into other spaces in their immediate school world (e.g., see how math relates to their walks in the hallway or work in the gym)?

2. Ground and build math concepts into known objects for children. When introducing, extending or innovating a math idea is it organic and natural to the child's explorations and world? For example, it is more natural to engage children in an exploration of measurement if they measure things in their immediate world using the stick they are playing with, instead of using a standardized ruler (e.g., "Can you find something the same length as your stick in the yard?", "What is taller than your body?").

3. Use available math moments with children. In our classroom we have a large block of uninterrupted play each day. It's sometimes challenging to manage children, materials and activities during center time. However I try to engage with children as much as possible in the activities, and take on the role of 'play partner' together with them. When I am actively playing I am able to closely observe what they are saying and doing, helping me to identify and extend the rich math learning that is organically occurring (e.g., helping children to recognize why their tower keeps falling, using math terms when they equally share the play dough, introducing math terms as they discuss how many cars are in their parking lot).

4. Become a math role-model for children, families, and colleagues. Even if math isn't your favourite subject, how do you discuss it within your school and classroom? Are you excited by new activities and resources? Do you demonstrate a growth mindset? When mathematical situations arise with children that you aren't sure of, can you use these opportunities to showcase positive thinking and problem-solving? Share your new math learning with others - suggest articles and books you're reading and post these throughout your classroom to enhance documentation displays.

5. Find the math in everything. Many educators plan forward by choosing curriculum and programming expectations and then building activities to fulfill these. Try back-mapping activities from time to time; embrace child-centered, organic experiences and then deconstruct them in order to identify the rich math concepts and curriculum expectations that they utilize. You'll be surprised to find that math happens in almost every experience children have in the classroom.

6. Try looking at life through a mathematical lens. When planning invitations for learning in your classroom, see what math you can sneak in as well. Changing one or two elements of the experience might be enough to engage children in rich math. It reminds me of how I used to sneak veggies into my children's meals - a little can go a long way!

7. Collaborate mathematically with colleagues. Share new ideas and resources informally. It's easy and effective to create math invitations and activities and share these within your school or division. If every educator plans one or two activities and these are shared, children will benefit from many rich and interesting games and activities without the burden of planning and preparation it would take one educator to accomplish the same.

8. Record and celebrate your math moments. Help children, families, and colleagues recognize that math happens everywhere in the classroom by creating a documentation display with photos, anecdotal observations and connections to curriculum. This bulletin board can be built over the course of the school year as artifacts of learning are continually added by staff and students. Keep sticky notes nearby and invite observers to record their own ideas and share them by posting the notes within the documentation.

9. Engage families in joyful math with children outside of school. Consider ways that you can promote and extend math for children after school. Encouraging families to play math games and activities together with their children will not only provide children with additional meaningful math moments, but it may help older family members reconcile their fear or dislike of math. In our classroom we send home 'family math bags' once a week. These are filled with math invitations and materials so that children and their families can play games inspired by our classroom work.

10. Build your collection of math stories, songs and games. Children love to sing, dance and play games. Ask colleagues to share their favourite games and activities, and use these to help with transitions and other 'math moments' throughout the day. Quite often the words in songs can easily be improvised to match something happening in your classroom, and many rich storybooks have mathematical elements and problems embedded within them that can inspire children.

These are just a few of the ways in which I cultivate a rich math space with children. I'd love to hear from you! Share your math ideas in the comments below, and consider following my Facebook page  Joyful Learning in the Early Years for regular math ideas from my classroom!

Sunday, June 16, 2019

What does the Equal Sign Really Mean?

A few years ago I was presented with this question at a workshop and asked to consider how a group of students would respond. After some discussion our group thought that children might recognize the third as incorrect. I was surprised when the presenter shared that most children, regardless of grade or age, think that the third example is right and the rest are incorrect. Why is that?

Most children associate the equal sign (=) with the word 'answer', so they look for traditional algebraic representations. This is why despite the third statement being incorrect, it looks like something most children regularly see and use in math (addend plus addend equals sum) so they assume it is correct. Example one, two and four don't look typical for many children, so they aren't sure about them and assume they are incorrect.

Mathematical equations must have an equal sign indicating that two expressions have the same value.

10 is the same (or equal to) 10.
3 + 7 and 8 + 2 both equal 10.
10 is the same as 9 + 1.

Therefore providing children introductions to many different ways of writing equations, and tangible, hands-on experience with the idea of equality is very important. Despite their age young children are capable of using complex math in context, including the proper terminology for symbols like the equal sign. This can be done in many ways in kindergarten. Over the course of the last year we have been on a journey to help children understand the equal sign. This blog post outlines some of what we have done in order to achieve success.

Using Proper Terminology in Math Discussion and Discourse

In our classroom we start each morning with a number talk during our morning message. I was curious to see what my students knew about the equal sign, and how they would describe their thinking mathematically. I asked the following question:

The first few times we reviewed this prompt children replied by telling me that the equal sign meant 'answer'. They knew that four added to one was the same as five, but they could not articulate this clearly. I knew that much practice was needed to help children look at numbers in new ways, explore the idea of equal amounts and equality, and play with equations in different ways. After a few months exploring these throughout many whole and small group conversations and math invitations, I again asked children the same question. They responded by saying:

"Four plus one is the same as five. They are both five."
"Both sides are the same. They are equal. It's like if I gave you four and one cookies and I had five. We would both have the same. It would be fair."
"Each side is the same as the other."
"It's like this." (child holds up one hand and shows five fingers and then holds up the other hand with five fingers. "Each hand has five fingers. They are the same."

In our classroom we explored the following activities many times and in many different contexts to help build this algebraic understanding. Much of this work happened before we even looked at a written equation with numbers and symbols.

Building on Mirrors

We offered children mini wooden cubes on mirrors. As they built towers children realized that the reflections of their creations were doubling the total amount that they used. This was a rich opportunity to discuss the idea of equal (e.g., "The number of blocks used in your tower is equal to the blocks in its reflection.") and doubling (e.g., "We can double the number of blocks you used in your tower to calculate the total number of blocks used.").
Subitizing Match

Understanding that there are many different ways to represent the same, or equal amounts, is a foundational number sense strategy. Not only does this help children become more accurate and confident when identifying/calculating/comparing sets of objects, it provides an opportunity to show equal amounts (e.g., five tallies = five dots = the number 5).

We used subitizing cards and encouraged children to match them to random numbers written on a chart...
 ...and also played games where children were encouraged to find similarities between number representations. In this game children were asked to find at least three different dot arrangements to represent the same number.

Creating equal number strings is also helpful. Sometimes we will use the date as a number prompt and ask the children to explore representations in different ways. In the following picture we showed children three different ways of arranging 8 hearts and asked them to create equations based on what they saw. 

After exploring the arrangements the children shared the following:

4 = 4 
4 + 4 = 8
2 + 2 + 2 + 2 = 8
8 + 0 = 8
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 8

Sharing at Centers

Play dough is one of the most popular centers in our classroom and despite making a double batch each week, sharing equally among the large number of friends tends to be problematic. Usually the first child to visit the center grabs the entire big ball of dough, and reluctantly tears off small amounts to the children who slowly make their way there. "That's not fair! You have more than me!" is frequently heard at the table. As educators we felt this would be an excellent, real life situation to help children think about equality and equal amounts in a meaningful context. In addition to supporting and scaffolding this directly at the center, we encouraged children to think about fair, equal amounts by adding plates and laminated photos of each child to the center.    
We also added tools like cutters and asked children how they could divide the play dough into equal amounts. "How do you know it's equal?" was an interesting conversation starter and the children's ideas for equality were interesting (e.g., "We could see if it fits in the same container.", "We could measure it with a scale to see if it weighs the same.")  
Real Life Math Problems

Inspired by the children's problem solving at the play dough center, we used our morning message to ask children deeper, more complex questions regarding equality and fair sharing. Because they love seeing themselves on the morning message, it was effective to ask how four children might share six cookies equally. 
The children saw the cookies as two groups of three, and then split the three cookies in three ways. We used arrows to represent what it was they were saying. After some conversation and use of real props they also recognized that three halves were equal to one whole and one half of a cookie, helping us delve in early fraction work.

1/2 + 1/2 + 1/2 = 1 + 1/2 

  Building Equals

The children love to create with pattern blocks and are especially skilled at designing intricate tessellations. Wanting to introduce the concept of rotational symmetry we created eight equal sections using tape on the carpet. After building the children recognized that the blocks used in each section were equal to, or the same, as the other seven. This activity was also made available during outdoor play by placing tape directly on the pavement and offering children a basket of pattern blocks.  
  Equation Clothesline

Another equality invitation offered to children for exploration was the equation clothesline. This consisted of a string hung between two posts, clothespins, and subitizing cards. Children were able to represent balanced equations by finding different representations of numbers and pinning them on the clothesline. Equations could be simple (as shown in the photo where the number 2 = 2 dots on a five frame) or complex by adding addition or subtraction signs on each side of the equation (1 + 1 = 3 - 1). This provided multiple entry points into the activity with a way of differentiating it for children's needs and interests. 
Visual Representation of an Equation

In order to help children move towards thinking in algebraic representations we provided a visual 'scale' along with numbers written on sticky notes. Children were invited to try and create a balanced equation by first placing numbers on the scale and then adding the equal sign (also written on a sticky note) to the visual. We did not write the equal sign directly on the paper because we also had 'greater than' and 'less than' signs offered on stickies in order to differentiate the activity for children who were working at that level. 
Equation Sort

Another activity to help children practise identifying correct equations is the 'equation sort'. In this activity we presented children with a number of equations and asked them to sort them as 'correct' or 'incorrect'. Children had to justify their choices by using math language to describe their thinking. A great extension is to offer children blank strips and encourage them to create their own correct and incorrect equations and place them under the corresponding categories.

Greater Than, Less Than

Now that children are comfortable with the concept of equality, we are exploring other relationships that numbers have with one another. Inequality is something that children have expressed an interest it. We are starting our explorations by using the language 'less than' and 'greater than' in contexts to describe sets of objects, and creating the  <  and  >  symbols to show these relationships. Cubes and craft sticks are an easy invitation to try out.

It will be interesting to see where this journey continues. Please feel free to share your explorations and activities with equality and inequality in the comments below, or tweet us your ideas and feedback at @McLennan1977.
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