Building Algebraic Thinking: Understanding the Equal Sign

Algebraic thinking starts with making sense of relationships: noticing patterns, explaining why they work, and using those relationships to reason beyond a single calculation. Understanding the equal sign is one foundation for that work. A student can add correctly and still misunderstand what the equal sign tells us about two quantities.

Consider 4 + 3 = ___ + 2. A student who adds all three visible numbers gets 9. The addition is correct, but 9 doesn’t make this equation true. Before correcting the response, we have a more useful question to explore: What does the student think the equal sign is telling them?

The number alone does not tell us how a student reasoned. Ask, “How did you decide?”

This week’s focus, 1.OA.D.7, asks first graders to understand the meaning of the equal sign and decide whether addition and subtraction equations are true or false. In this week’s Math Standard Monday video, I explore a common misunderstanding of the equal sign and how connecting cubes can help clarify its meaning. Explaining why two expressions have the same value builds on foundations that begin in kindergarten.

Build Foundations for Algebraic Thinking in Kindergarten

In kindergarten, children put quantities together and take them apart. K.OA.A.3 asks them to find different pairs of parts for a number within 10. Counters, cubes, and drawings help them see that a whole can be decomposed in more than one way.

Start with 5 counters and separate them into 2 and 3. Writing 5 = 2 + 3 highlights the known whole and how it has been decomposed. Keep those 5 counters in view and change the groups to 4 and 1. Ask, “What changed? What stayed the same?” The parts are different, but no counters were added or removed. Both arrangements represent the same whole.

Composing starts with the parts. Give children groups of 2 and 3 and ask how many there are altogether. Writing 2 + 3 = 5 records composing, or putting those parts together. Whether children begin with a whole and explore its parts or begin with parts and find the whole, the equal sign connects quantities with the same value.

Equations record the thinking children are doing with quantities. Kindergarten children should encounter them; writing equations is encouraged, not required. Grade 1 builds on those experiences by making understanding the equal sign an explicit expectation.

Start with known parts to compose a whole, or start with a known whole to explore its parts.

For a closer look at this foundation, revisit my video K.OA.A.3 - Decomposition and the Equal Sign.

Connect the Language to the Meaning

Seeing = or hearing “equals” does not automatically give children an understanding of equality. That meaning develops as they compare quantities and explain how they know the amounts are equal, even when the objects are arranged differently.

Reading = as “is the same as” can connect those experiences to the symbol. The goal is more than learning a phrase: students need to understand that quantities can have the same value even when they look different or are represented in different ways. Connect “is the same as” to “equals” as that understanding grows, so students can use both expressions meaningfully.

For example, read 4 + 3 = ___ + 2 as “4 + 3 is the same as what + 2?” Now the question is about a relationship, not a signal to add every number. Ask students to model what would make the two sides the same.

Standards note: This four-position prompt opens a conversation about equality; it goes beyond the three-whole-number equation form specified in 1.OA.D.8.

My video The Equal Sign in Kindergarten: Banned Words vs. Bridge Words explores this language connection. But changing our words is only a starting point. The research helps explain how connecting models, notation, and discussion supports algebraic thinking.

What the Research Helps Us See

Correct Calculation is not the Same as Understanding Equality

In Children’s Understanding of Equality: A Foundation for Algebra, Falkner, Levi, and Carpenter describe responses to 8 + 4 = ___ + 5. Students offered 12 or 17—calculations using the visible numbers rather than a value that makes the equation true. The challenge was not simply adding correctly; it was interpreting what the equation said.

A kindergarten example makes the issue especially clear. Children chose 9 for 4 + 5 = ___ + 6. Their teacher built stacks of 4 and 5 cubes, then stacks of 9 and 6. Children recognized that the collections were unequal and could suggest how to make them equal. Yet they still believed 9 belonged in the written blank.

Standards note: This was a research task used to investigate children’s thinking, not a kindergarten expectation. Finding a missing number in this equation goes beyond the kindergarten standards; the example illustrates the difference between understanding quantities and interpreting symbols.

That is an important gap in developing algebraic thinking—recognizing a relationship with objects does not guarantee interpreting its symbolic representation. In our True or False routine, we can pause to connect each part of the model to each part of the equation. Ask students to show what the two sides represent and explain why their model supports their claim. The materials are a way into that conversation, not a substitute for it.

Illustration of the classroom example described by Falkner, Levi, and Carpenter (1999). Recognizing unequal amounts did not automatically change how these children interpreted the equation.

Discussion Can Change What Students Attend To

Molina and Ambrose began with the same equation, 8 + 4 = ___ + 5, in their third-grade classroom. They worked with 18 students over five sessions, with attendance varying. Students initially proposed 12, then 17 when attention was drawn to the 5. Explaining the conventional meaning of = did not settle the issue: misconceptions remained at the next session.

The researchers then used varied true-or-false equations, discussion, and opportunities to correct false statements and create equations. Later, students explained that 15 + 2 = 15 + 3 was false by comparing the second parts, rather than calculating both totals. They also described compensation in 51 + 51 = 50 + 52: moving one from one part to the other preserves the whole.

Progress was uneven. Some students continued calculating, and some interpretations remained fragile. This small classroom study is not a guarantee for every learner, but it shows why the explanation matters. A recurring True or False routine gives students opportunities to compare ideas, defend a claim, and revisit the meaning of = across different equations—not just repeat its definition.

Develop Algebraic Thinking with a True or False Routine

In a True or False routine, students consider an equation, decide whether it is true, and justify their decision. Unlike a quick answer check, the discussion focuses on how they know. Carefully chosen equations invite students to notice structure: a part that stays the same, a part that changes, or a change in one part balanced by a change in another. This is an opportunity to engage in SMP 7: Look for and make use of structure—not a race to calculate.

This week’s version focuses on equality with addition. Reveal one equation at a time and give students time to think before inviting explanations. The goal is to notice relationships without needing to calculate both totals before making a decision.

Connecting cubes are a starting point for making the relationships visible. The first set shows one equation per slide, with larger cube models beneath it. Connect each part of the model to the written expression.

Build an annotated class poster during the discussion, recording equations alongside students’ explanations, sketches, and cube models. The poster keeps earlier examples available for comparison while each supported slide focuses on one equation.

Try the second set the next day or later. It uses new equations with related structures, without pictured cubes, and reveals them cumulatively so earlier equations remain visible. Ask students to explain how they know—not just give a true-or-false response. Can they carry the reasoning from the models into the notation?

If that connection is not yet clear, return to the cubes and link the model to the equation again. Keep physical cubes available as needed. The timing is flexible; the purpose is to learn what students understand, not to remove support on a schedule.

Get the free activity and use students’ explanations to decide what to explore next.

A correct true-or-false response is only the beginning. Listen for whether students are calculating, comparing quantities, or using relationships.

How Understanding Equality Supports Fluency

The relationships students explain in a True or False routine also help them use known facts to find unknown facts. If 4 + 4 = 8, what happens when one part changes from 4 to 3? The other part stays the same, so the total decreases by 1: 4 + 3 = 8 − 1. Connecting cubes can make that change visible.

Keep one part unchanged and decrease the other by 1: the whole is 1 less. A known fact helps explain a new one.

This is the connection explored in my earlier work, Doubles Math Facts: From Ten Frames to Near Doubles. Students do not have to start over with every calculation. They can use a relationship they understand to reason about something new.

That goes beyond completing fact-family houses or listing related facts. Those activities can record relationships, but the important question is whether students can explain them. This is where algebraic thinking supports fluency: understanding equality helps turn individual facts into connected mathematical knowledge. Students learn to ask what changed, what stayed the same, and how one relationship can help them understand another.

Trace the Learning Across Grades

These foundations of algebraic thinking matter beyond first-grade fluency. This month, we’ll trace how they develop across grades: from understanding equality to finding unknowns, recognizing equivalent expressions, and writing equations to solve problems. Seeing those connections helps us identify what students are building on and which understanding needs attention next.

Ready to make these connections in your own planning? Join us for the January 2027 Standards Summits to dig into what your grade-level standards require, trace the learning that comes before and after, and consider what those connections mean for instruction.

• K–2: January 19

• Grades 3–5: January 20

• Grades 6–8: January 21

Bring your questions, invite a colleague, and make time to study the mathematics behind the standards.

Choose your grade band and register for a Standards Summit.