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.

Sunday, September 30, 2018

Puddle Play - Rethinking the 'Math Classroom'

We are always looking for opportunities to head outside and enjoy the weather, even when it might be a little wet or cold. Last week our area enjoyed quite a bit of heavy autumn rain. The children spent much of the morning peering through the window and marveling at how much rain was falling. After a few hours they noticed a giant puddle forming in the corner. As the rain continued to fall and the puddle grew, the children became concerned that their outdoor play time would be hampered by the rain.

Fortunately by the early afternoon the rain tapered and the children were able to head outdoors. We are lucky to have a class set of rain boots purchased by the school parent council for use in our outdoor classroom. This way children can enjoy playing in our school yard, even when they might forget to bring their boots to school that day. We are always prepared and the children ready to explore the world outdoors. Even the giant puddle would be no problem that day!

The children scurried outside after eating lunch - many wearing mismatched school boots because they couldn't get them on their feet fast enough because they were so eager to explore the water.


As I approached them I was amazed by the authentic math talk developing in this natural and authentic learning environment. As I listened into their conversations I overhead children wondering with each other about amazingly big math ideas...

How many children could fit in the puddle?  Did each person's feet have to be touching to be counted? Did children who were jumping in and out of the puddle count too?

How deep was the water? How much water was there really in the big puddle? Could it fill a bath tub? How could it even be measured?

Who could make the biggest splash? How would they even judge how big the splash was? Could it be measured? Was the biggest splash the one that soaked the most children standing nearby?

Could the water be used to make soup? How much water was needed in the recipe? Would anyone even want to eat mud soup?

How much water was in one's rain boots after a big splash? How long did it take to dump it out? How did all that water get in there in the first place?

How much more water was needed to cover the stump? Was the stump floating or sinking in the water?  How tall was the stump? What if it rained more...would the stump be under water?

How long would it take to run across the puddle? Who could run the fastest over the water? How could we measure and record the puddle races?

The children were making connections between their puddle play and math in the world around them. The questions they were posing about their experiences in the water were meaningful to them, supporting and strengthening their productive disposition towards math. As an educator involved in their play, I was able to listen to their questions and facilitate conversation and critical thinking about the big math ideas. How could we figure out who could make the biggest splash? What experiences did the children have measuring the size of something irregular. What tools and resources were available to help support this inquiry? Could technology play a role? Would children be interested in revisiting these math questions at a later time or would their interest only occur when playing in the puddles?

As the children and I engaged in conversations about their questions they were developing adaptive reasoning skills - this is the capacity for logical thought, reflection and justification in their math thinking. As children connected what they were observing and experiencing in the puddle play to their own unique experiences and ideas, they were engaged in rich learning as they reflected upon and justified their questions, ideas and strategies to solve the puddle math problems.

Even though many of the children's questions were not answered, the purposeful outdoor math exploration encouraged children to develop a strong conceptual understanding of a variety of developmentally appropriate math topics related specifically to our curriculum including measurement, counting, capacity, classification, time and quantity. I was able to support their conversations and provide suggestions and strategies in the moment. I became a play participant together with them by playing in the puddles myself.

What had originally looked to be a damper on our outdoor fun turned into a complex and layered opportunity for rich math thinking during an activity that most children love to do - explore the rain. It just shows that math can happen anywhere, anytime, when we are willing to rethink what the 'math classroom' should look like.

Sunday, September 16, 2018

Anchor of Five

Anchor of 5

A successful math program for children will have an emphasis on number sense as its foundation. Number sense is a natural part of all other strands (e.g., geometry, patterning, data management). Exploring number relationships help children build fluency, accuracy and confidence. Five frames provide a visual reference to the anchor of 5. Five is a 'friendly number' for children. They associate five with the most natural of math 'manipulatives' that they always have available...fingers on one hand! The number system that we use in Canada encourages an understanding of place value that is dependent on groupings of 10, and understanding groups of 5 will evolve into 10. This is a key foundation for future place value work. Here is a review of some of our math work this week. 
Read alouds
We used many engaging, patterned texts during our whole group circle time that focused on groups of 5. In books like 'Five Busy Beavers' a group of 5 beavers slowly decreases to 1 as each beaver leaves the water for other adventures. Children can see the group decrease by 1 each time, and predict what the new number will be. They can subitize the new number as they observe the number of beavers on each page, or follow along and use their fingers to chant along with the text. This book can then be added to a math centre where manipulatives can be provided to further enhance the text and encourage children to play with the numbers 1 through 5.
A Number Station
During free choice time the children had the opportunity to visit a math centre where various manipulatives and tools were made available for children to play with the numbers. A number line, wooden and mirror numbers, five frames, finger tracers, and natural materials were available for children to explore. Students matched, counted, sorted, patterned, and ordered the manipulatives, often composing groups of 5.
Morning Message
Each morning we start our day with a morning message. One of the most important words that children first learn to read and write are their names (their own, and those of their peers). We used our 'star of the day' to model how our names fit into five frames (and sometimes beyond the five frame if the name has more than five letters). This helped us conceptualize the anchor of five and also introduced some concepts of print too (e.g., that words are composed of letters and that letters represent sounds).
Number Line
We brought a number line outdoors with us during our outdoor play time. It was interesting to see how the children created their own games without adult prompting. Some children gathered natural materials and placed them next to the numbers (e.g., 7 stones next to the number 7). Others used the line as a tool in a jumping game, starting at the 0 and seeing who could jump the furthest and reach the biggest number!
How Many are Hiding?
Whole group time is also a great opportunity to introduce meaningful math games that children can then play in small groups or during free choice play time. To help children compose to the anchor of five, we used a group of five unifix cubes. Children are first shown the five in a line. The player then hides some cubes behind his/her back and shows the group the remaining cubes. The group has to calculate how many cubes are hidden, encouraging them to subitize and compose to the anchor of 5. They indicate the missing quantity by holding their fingers up to the player who then reveals the missing quantity.


We love to sing each day. Our children loved the fingerplay "Five Little Monkeys Swinging in a Tree". To enrich the experience with meaningful math, we added a magnetic five frame to the song. As the children sang along and used their fingers to decompose the number 5 to 1, we removed counters from the frame as they count. During play time many children enjoyed leading their peers in a singing of the song!

Pentomino Challenge
Pentominoes are a wonderful math manipulative that encourage spatial reasoning and use an anchor of five as each unique piece is created using five small squares. A challenge that encouraged perseverance and spatial logic this week involved challenging the children to fill a standard cookie tray with pieces, leaving no gaps in the puzzle. 

We love learning from others! Share your favourite math activities that encourage an exploration of the anchor of 5 in the comments below!

Thursday, May 10, 2018

Printable Pentominoes

In our kindergarten classroom children enjoy spending time working with math manipulatives 
that encourage playful explorations with shape in a variety of ways. One of our favourite tools are 
versatile pentominoes.

Pentominoes are polygons made of five, equal-sized squares connected edge-to-edge. There are twelve 
different pentominoes in one set. You can purchase pentominoes from educational resource stores, or 
print paper copies on card stock and laminate for children’s use here:

There are many reasons why pentominoes are an essential math tool for any early childhood classroom. 

1. are gamelike in nature and promote a positive attitude towards math
2. encourage cooperation and collaboration among children 
3. Promote math thinking in a variety of areas including spatial reasoning (logic when solving
 puzzles, symmetry, reflection, rotation, design), measurement (considering the area and perimeter 
of designs), and number sense (counting the number of tiles or squares in a design, calculating the 
total number of squares using the anchor of 5)
 We have used pentominoes in many classroom activities. To see what we've done and follow our math journey, follow us on twitter @McLennan1977. Share your favourite pentomino activities in the comment section below!

Wednesday, May 9, 2018

Flipped Hundred Chart

Helping children conceptualize numbers and find meaningful ways to think about their relationships is a goal in our kindergarten classroom. We encourage problem solving and working with number strategies in daily number talks, and are always on the lookout for interesting ways to compliment the rich math learning opportunities we are observing in the children's play.

Recently I became a member of the National Council of Teachers of Math (NCTM A highlight of membership is that I have access to the journal Teaching Children Mathematics, which is filled with research-based articles outlining interesting and developmentally appropriate practice for the early years. In the December 2017 (24, 3) issue there is a fascinating article by Jennifer M. Bay-Williams and Graham Fletcher called A Bottom-Up Hundred Chart? In this piece the authors challenge educators to consider the potential for enriching children's learning if the popular math tool is flipped upside down.

Bay-Williams and Fletcher share that a flipped hundred chart makes sense because when children use the chart to solve equations, the language they use to describe direction on the chart matches their understanding of the operation - if adding the number appears to get taller, bigger and greater as physically modeled when children track the addition sentence by moving up on the chart and if subtracting the number appears to shrink, moving downwards and getting smaller by descending the chart (e.g., if a child is solving 13 + 12 s/he would first point to the thirteen, move upward one space and then to the right two). The authors also suggest a number of different activities for exploring the chart including cutting one into a 'number puzzle', encouraging children to find mystery numbers, and assigning children a number and challenging them to find all the number's 'neighbours'. These activities are great ways for children to physically and mentally manipulate the chart, helping them to become more comfortable working with the numbers.

After reflecting upon our math program I have created five additional activities that I believe will continue to challenge children and encourage them to strengthen their understanding and confidence working with the numbers 1 - 100.

Guess my number!

Display a flipped hundred chart and mentally think of a number that the children will have to guess. Give children one clue at a time to help guide them to your number (e.g., my number is less than 50, my number doesn't have a 6 in it, my number is odd). The children can consider the clues and cross out the numbers that don't follow the clues. With additional clues more numbers will be crossed off the chart until children guess the correct one. Reverse roles and invite the children to think of a number and give the clues to you!

Pentomino Trace

Our children love to manipulate pentominoes - using them to fill frames and trays and solving intricate puzzles. Children can also be encouraged to use them as tracers, matching them to the chart and outlining the numbers contained within. Once a collection of numbers has been traced, encourage children to find something that all numbers have in common (e.g., they each have a 4 in them; they are all greater than 36). Tip - ensure your flipped hundred chart is printed to the same size as your pentomino set to ensure an accurate match.

Dry Erase Number Write

Our children love to write with dry erase markers. Create a variety of 'missing number' flipped charts, laminate, and invite the children to practise filling in the missing numbers.

Flipped Hundred Chart Coding

We spend a lot of time coding in our classroom and see potential for incorporating coding directions and spatial reasoning into exploring the flipped hundred chart. Display the chart and determine a starting number. Provide children with a series of verbal or written coding directions as they move from the starting number and reach the end number. Once the final number is determined encourage children to create an statement or equation that describes the relationship between the two numbers.

Start at 3.
Move up 4.
Move right 5.
The number is 48!

 What do you know about 3 and 48? (48 - 3 = 45, 3 + 45 = 48, 3 + 5 + 40 = 48)

Roll, Subtract and Race!

Use an enlarged copy of the flipped hundred chart as a board game. Two (or more players) can use recycled board game pieces and place them on the starting point 100 spot. Each player takes a turn rolling one die and moving their player backwards on the chart (or 2 to make the game more difficult). Students can be encouraged to think in equations (If I start at 100 and roll 5, 100 - 5 places me on the 95 spot). First player to move off the board into the 0 spot wins!

What other ways can children learn with the flipped hundred chart? 
Feel free to share other ideas in the comment section below or tweet #flipped100schart!
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