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.

What Does Equal Really Mean? Exploring the concept of equivalency in guided and playful explorations

I had the pleasure of attending a workshop presented by Dr. Alex Lawson at the ETFO Math Conference this past weekend. In addition to being a renowned expert in Ontario Mathematics I found Dr. Lawson to be a highly engaging and motivating speaker. Her ideas for enriching the mathematical space in the classroom and engaging children in higher level activities that embed a deeper understanding of and accuracy in computation were highly relevant. Dr. Lawson reminded me of the need for both child-guided exploration and explicit instruction when considering number sense in the classroom. 

Dr. Lawson spoke specifically about the recent interest and discussion that has been explored in the press regarding the debate between children needing to memorize their basic facts and guided exploration leading to number mastery. For those interested she has written a Ministry  monograph delving more deeply into this issue. I found Dr. Lawson's discussion of particular interest because of the ongoing debate in FDK regarding just how much explicit instruction and guidance children should have in a playful, emergent program. Many teachers wonder about the balance between guided instruction and open exploration. How might a teacher initiate and introduce a math concept that has not yet emerged in the children's play? What is the link between explicit instruction and small group practice? How can educators embed more meaningful math explorations into an emergent space? What strategies can educators use to ensure all children have access to robust math explorations in meaningful ways?

Dr. Lawson spoke specifically about the need for children to have rich, meaningful interactions within the classroom that promote an internalization of number sense, specifically computation, fluency, and accuracy. Memorizing facts and engaging in rote, didactic activities may help a child recall facts, but there won't be a deep understanding or flexibility when working with numbers and children won't necessarily know the 'why' behind the math.

I was fascinated by the following slide that Dr. Lawson shared. When asked which equation was wrong, most children felt all were wrong except for the third. Why? They rationalized that it was the only equation written in what they perceived to be the correct format. The children were thinking that the equal sign meant 'answer' and had not really internalized the concept that the equal sign represents that each side is the same as the other. They had memorized what they felt was the correct way to write an equation without understanding how the symbols and numbers worked together. This was troubling for me.

When I returned to the classroom I wondered what my children would think if I presented the same question to them. I wrote four different equations and asked them which one was wrong.

I was so pleased when many of them immediately pointed out the third as being incorrect. When I asked why, they automatically totaled the different addition ideas on each side of the equation and stated that both sides needed to be the same.

"2 plus 1 equals 3 [is wrong]. It would have to be 2 plus 3 to equal 5!"

"The other ones all have the same numbers. 3 plus 1 is the same as 1 plus 3."

"The last one is 6 for both sides."

As they explained their thinking I wrote their ideas directly onto our morning message.



I was pleased that many recognized the incorrect equation but wanted to make sure as many children as possible had entry points into thinking about the idea of equivalency and knowing that each side of an equal sign needs to be the same in order for the statement to be correct. I asked the children to sit around the edge of the circle and placed a set of subitizing cards in the middle. The cards show different ways of representing numbers (numerals, dots, ten frames, cubes, fingers, etc.).

In our pocket chart I placed a number of equal signs and asked the children to consider how they might organize the subitizing cards. A number of children demonstrated how they could use numbers in different ways and placed the equal sign in the middle to show these were the same.

This generated a lot of interesting conversation as children debated the numbers. I'm hoping to revisit this idea in another few days and encourage the children to add addition or subtraction signs on either side of the equal sign so they are composing and decomposing numbers yet making each side balanced and equal to the other.

This activity was available for the children to revisit and explore during our next large play blocks. Many children wrote their own equal signs and used these as well.

In the above example I introduced the idea of equal numbers. This was a teacher-initiated activity however I also wanted to provide an opportunity for the children to have open exploration so I created an invitation for them to explore the concept of balancing and equal measurements using loose parts and a scale.

Many children were drawn to the interesting materials. They were curious about what the scales were and how they worked and spent much time with open ended exploration.

"Look! I know when they are equal! These arrows need to be in the same place!"

As the children manipulated the loose parts on the scale they noticed that the heavier side was lower than the lighter side and the arrows did not line up. In order for both sides to be equal the scale had to be level and then the arrows would touch. 

After open ended exploration with this concept the children shared their thinking in the next whole group circle. We named the learning and shared math terminology to help the children engage in meaningful conversations the next time they visited the materials - balance, level, and weight were especially important math words to hear and learn.

We began to notice an interesting thing during the next play block. Many children were exploring the idea of being 'equal' using different play props and ways of measuring materials.

We noticed measurement at the playdough centre... 

"2 parts can come together to be a whole."

"I have equal slices of pizza so all the friends have the same to eat."

"4 equal slices of pie to eat too!"

It was so interesting to see how the children interpreted our initial conversations of what 'equal' meant and applied their understandings to new and interesting play situations. We heard many conversations today and noted that some children were beginning to use language that described fractions - this is especially interesting as fractions are one of the topics most children have difficulties with. If kindergarten children are becoming comfortable using rich math language in the appropriate corresponding context, their mathematical mindsets will flourish as they see themselves as capable and invested mathematicians in the classroom and beyond!