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What does it really mean to understand the equal sign? For many students, this tiny symbol carries a big misunderstanding. It is often seen as a cue to calculate rather than a signal of relationship or balance. That misconception may seem small, but it can have lasting effects on how students make sense of equations and build readiness for algebra.
This episode explores how early experiences shape our students’ understanding of equality. We will look at how small shifts in classroom routines, language, and materials can create clarity and deeper reasoning.
This conversation is for anyone who has ever been surprised by how complex the equal sign can be for young learners and is ready to help build a deeper understanding from the very beginning.
Links mentioned in this episode:
Understanding the Equal Sign in Early Math
Hello again, Meaning-Makers!
Today, we’re going to tackle the topic of the equal sign. We will discuss the meaning of the equal sign, as well as the misconceptions students have about the symbol.
We are going to explore and answer the questions, “What exactly is the meaning of the equal sign?”, “What do students traditionally believe to be true about the equal sign?”, “Why do students have these misconceptions?” and finally, “What can I do about it?”
What Does the Equal Sign Really Mean?
First, let’s unpack the true meaning of the equal sign. The equal sign is a relational symbol that represents balance, or equivalence.
In the simplest of terms, the phrase, “has the same value as” can be replaced with the equal sign in any equation, and this would describe the context of the equation, every time.
The Common Core Standards call for first graders to understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. Students are also expected to determine the unknown whole number in an addition or subtraction equation relating three whole numbers. These standards are meant to set a firm foundation for later learning of algebra.
A Surprising Look at Student Misconceptions About the Equal Sign
I want you to stop what you’re doing and write down this equation: (If you’re unable to write it down, try to picture it in your head.) 8 + 4 = ___ + 5.
Now imagine you present this equation to your students. How would you want your students to solve this? What would you expect their answers to be?
A group of fifteen teachers and three researchers affiliated with the Wisconsin Center for Educational Research posed this same equation to more than 700 first through sixth-graders, and collected the data of the responses.
Does it surprise you that only four percent of first and second graders identified the correct solution as 7? One would assume that the older students had more success with this, but only three percent of third through fifth graders answered correctly. What is probably the most shocking part about this data is that zero percent of sixth-graders arrived at the correct solution.
What do you suppose the most common answer was to this problem? Well, more than half of the students in the study believed the solution to be 12, and nearly twenty percent of students thought the answer was 17. What does this data reveal about students’ perceptions of the equal sign?
Why Students Misunderstand the Equal Sign
Conceptualizing the true and accurate meaning of the equal sign is a lot more complex than one would think.
In Kindergarten, students are introduced to the plus, minus, and equal symbols. Young learners begin to notice a pattern that “plus” and “minus” tells them to do something, so it only makes sense that students also begin to internalize that the equal sign is also an action symbol. In fact, no logical reason exists that the equals sign does not mean “compute”.
Common Misconceptions About the Equal Sign
A very common misconception that stems from this misunderstanding, and the misconception that is most evident in the study I previously mentioned, is that the equal sign means “the answer” to a number sentence. Students view the equal sign as a computational symbol rather than a relational symbol.
In other words, students believe that they have to “equals” a number sentence to get an answer, similar to if you were pushing buttons on a calculator.
Another common misconception is that the equal sign must always be placed at the end, or to the far right, of the equation. Many students are thrown off when they see an equal sign placed further on the left side of the equation and believe that it is backwards, or on the wrong side.
Lastly, students also begin to assume that only one number should come after the equal sign.
Generally speaking, most math curricula and the language used when referring to the equal sign do not support an accurate interpretation of equality, and students begin to make inaccurate conjectures very early on based on patterns they notice.
What Can Teachers Do to Support True Understanding of the Equal Sign?
So this begs the question, “What can I do to undo my students’ preconceived misconceptions about the equal sign, and how can I support them to develop a more accurate understanding of equality?”
If you listened to our previous episode, you heard me talk about the CRA approach. CRA stands for concrete, representational, and abstract. This approach emphasizes the importance of students exploring math concepts tangibly through the use of concrete manipulatives and connecting these understandings to a visual representation as well as abstractly, with the use of symbols.
I’m going to describe some examples of ways we can implement all three stages to support our students’ understanding of the equal sign.
Using Concrete Tools to Build Understanding About Equality
One of the most concrete and meaningful ways to represent equality is none other than a balance scale!
When using a balance scale to represent equations, it’s helpful to label the fulcrum of the scale with an equal sign. A simple sticky note will do the trick! This helps students internalize the purpose of the equal sign as representing balance.
Students can use a balance scale to determine if given equations are true or false. Any object can be used to represent values on the scale, but it’s important that the objects are the same in order for the exploration to work. I suggest using connecting cubes, or even counting bears if you have them.
For example, let’s say a student is trying to determine if the equation 5 + 2 = 3 + 4 is true or false. This can be modeled by putting 5 red cubes and 2 blue cubes on the left side of the scale and 3 green cubes and 4 yellow cubes on the right side. The different colors help students see the relationship to the different addends in the equation.
When five cubes and two cubes are placed on one side of the balance scale, and three cubes and four cubes are placed on the opposite side of the balance scale, the scale is perfectly horizontal which demonstrates that the equation is true! If the scale is imbalanced, then the equation it represents is false.
The balance scale can also be utilized to determine the unknown value in a true equation. Consider the problem 4 + 2 = ___ + 1. How could this be modeled and solved on a balance scale?
Well, four cubes and two cubes can be placed on the left side, and one cube can be placed on the right side of the scale. The scale will be visibly unbalanced at this stage.
A student can make the scale balanced by adding cubes on the right side of the scale until it is perfectly horizontal. Then count how many cubes were added to make the scale balanced!
Another constructive activity to explore equality is to build towers with connecting cubes. Similar to the balance scale activity, it is helpful for the students to represent each number in the equation with a different color.
Using the same example as before, I can determine if 5 + 2 has the same value as 3 + 4 by connecting 5 cubes of one color and 2 of another color together to form a tower and then connecting 3 cubes and 4 cubes together to form another tower. Place the towers side by side, and if the equation is true, the towers will have the same height!
Ten frames and counters can also be used to represent equations. Two ten frames can be placed side by side with each ten frame representing the values on both sides of the equal sign. The different addends can be visualized with red and yellow counters. With the ten frames side by side, students are able to compare them to determine if both ten frames represent the same value.
Using Pictorial Strategies and Visual Models to Represent Equality
Students can also represent equations with pictures or drawings. If students don’t have immediate access to an actual balance scale, they can draw one! They can draw circles or squares to represent the cubes or counters that would be placed on the balance scale in the concrete representation.
It would be helpful for students to color the shapes to represent the addends. If subtraction is involved, the students can show the removal of objects by simply crossing them out. Students may even choose to draw ten frames and counters. The fun thing about pictorial representations is that students can get creative. Any image really, can show a relationship of equivalence.
Encouraging Symbolic Thinking and Use of Precise Language
When a student is ready to demonstrate a concept abstractly, this means they can represent a math concept using numbers and symbols, or they can see the numbers and symbols and interpret the meaning of the symbols. Most of what students see on worksheets, textbooks, and assessments are in abstract form.
When it comes to the abstract phase, being mindful of our language in regard to equality, and the exposure we give our students is everything. This requires us to hold ourselves and our students accountable to shift our language so it always supports the real meaning of the equal sign.
I think it is also important to note that students can best understand equivalence when we use the language “same value as”, not “same as”. This is why: 4 + 6 equals 3 + 7, right? But if we say, 4 + 6 is the same as 3 + 7, that’s not necessarily true is it?
Imagine how confusing this is to a young learner when they see two different sets of numbers on either side of the equal sign, but here we are, telling them that they are the same. Simply adding “value” to the phrase, so it says “same value as,” can make a huge impact on a child’s ability to internalize the idea of equality.
Encourage your students to read equations as statements. Instead of saying, “Four plus three equals seven” they can read it as, “Four plus three has the same value as seven.”, or “Four plus three has a value of seven.”
Providing Exposure to a Variety of Equations
Exposure to a variety of equations is also just as important as language. The first grade standards that focus on equality refer to working with diversified equations including, but not limited to, the format of a = a (for example, 6 = 6), a + b = c + d (similar to 6 + 2 = 1 + 7), and c = a + b, in addition to the most traditional a + b = c format. It is crucial for students to see all of these formats, if we want them to squash the widely accepted idea that the equal sign means, “the answer.”
Thank you for joining me in this incredibly important discussion about the equal sign. I hope what we talked about today has left you feeling mindful about how a seemingly simple concept is actually quite complex for our students. And most of all, I hope you feel inspired to take these ideas back to your classroom. That’s all for today, Meaning-Makers. Have a great one!
