In this installment of our Unlocking Dyscalculia series, Adrianne Meldrum welcomes back Dr. Sarah Powell from UT Austin to talk about something deceptively simple: the equal sign.

And as it turns out, there’s a lot to know.

Table of Contents

A Symbol We Think We Understand

As Adrianne admits, many of us assume there isn’t much to unpack.

“When I was thinking about the questions, I thought, what is there really to know? And there is so much to know.”

Dr. Powell has spent nearly two decades researching how students understand the equal sign—and how misunderstandings about it can quietly derail math progress.

Her work began while developing Pirate Math, an intervention program focused on word problem solving. Students used equations like:

2 + 3 = _

2 + _ = 5

_+ 3 = 5

These equations represented the structure of word problems. But during her doctoral studies, Dr. Powell encountered research showing that many students misinterpret the equal sign.
That discovery changed everything.

A Small Symbol with Big Impact

Dr. Powell jokes that her family still teases her:

“Is she talking about the equal sign again?”

Nearly 20 years later, she is.

The equal sign is not just punctuation in math — it’s foundational. When students misunderstand it, the effects ripple outward into algebra readiness, equation solving, and higher-level reasoning.

For students with dyscalculia, especially, strengthening conceptual foundations isn’t optional — it’s essential.

What Does the Misconception Look Like?

Dr. Powell explains that many students, without explicit instruction, develop what researchers call an operational understanding of the equal sign.

Operational means:

“When I see this symbol, I need to do something.”

So if a student sees:

2 + 3 = _

They add and write 5. That works.

But the misunderstanding shows up when the format changes.

Consider:

2 + _ = 5

Many students will write 7 in the blank.

Why? Because they see:
🟣2

🔵5

🟢a plus sign

🟡an equal sign

And they’ve internalized the rule: When I see this, I combine the numbers.

The result?
They create a false statement:

2 + 7 = 5

This reveals that they are not thinking about balance — they’re thinking about computation.

Relational Thinking: What We Actually Want

Instead of viewing the equal sign as an instruction, students need to understand it as relational.

Relational means:

 
For example: 2 + 3 = 5

The left side (2 + 3) has the same value as the right side (5).

Dr. Powell emphasizes that students must think:

“This side is the same as that side.”

This shift is critical, especially as math becomes more complex.

When Equations Get More Complicated

Now consider:

2 + _ = 6 + 7

If a student still believes the equal sign means “do something,” they might:

 
But relational thinkers recognize:

This kind of reasoning depends on understanding balance, not just computation.

The research shows this misunderstanding persists far longer than we might expect.

Dr. Powell notes:
🔴Many middle school students still interpret the equal sign operationally.
🟠Some studies show that even college students default to “do something” thinking.

This isn’t a minor gap; it’s a foundational one.

The Powerful Connection to Greater Than and Less Than

One of the most fascinating insights Dr. Powell shares is how the equal sign connects directly to inequality symbols.

Visually:
➡️The equal sign is two parallel lines.

➡️If you “pinch” those lines on one side, you get a greater than or less than symbol.

Conceptually:
✅Equal sign → Both sides are the same.

✅Greater than → One side is a higher amount.

✅Less than → One side is a lower amount.

All three symbols describe relationships between two sides.

When students understand this family of symbols as relational, their number comparison skills strengthen alongside equation solving.

Making It Concrete: Why Manipulatives Matter

Adrianne highlights a powerful visual tool, a new balance-scale manipulative from hand2mind.

This tool physically tilts:
Balanced → shows an equal sign.

One side heavier → shows greater than.

The other side heavier → shows less than.

These kinds of visual and tactile supports are especially helpful for students with dyscalculia because they:

• Reduce abstraction
• Provide dual coding (visual + symbolic input)
• Reinforce relational meaning through experience

 
Traditional balance scales, linking cubes, and hands-on equation models can all support this shift in thinking.

Why This Matters for Dyscalculia

Students with dyscalculia often struggle with:
🔴Flexible thinking

🟠Holding relational information in working memory

🟡Recognizing patterns across formats

If the equal sign is misunderstood early, every increasingly complex equation builds on shaky ground.

But when students are explicitly taught that:

“The equal sign means the same as,”

They gain access to:

• Algebraic reasoning
• Equation solving
• Logical comparison
• Word problem structure
   
  It’s not about memorizing procedures.
  It’s about understanding relationships.
   

  What Parents and Teachers Can Watch For

  Here are signs a student may have an operational misunderstanding:
  ➡️They consistently add all numbers in an equation.

  ➡️They struggle when the unknown is not at the end.

  ➡️They get confused by equations like 7 = 3 + _.

  ➡️They rarely check whether both sides are equal.

  If you notice these patterns, it’s not a lack of effort; it’s likely a conceptual gap.

  And the good news?
  Conceptual gaps can be filled.

  So, when we introduce the equal sign, what mathematical language should we actually be using?

  Oftentimes, the shift begins with the words we choose.

  The Problem with “Equals Means Equals”

  Dr. Powell shares that when her team assesses students, they often begin with a simple question:

  “What does the equal sign mean to you?”

  The most common answer?

  “Equals.”

  When asked to explain further, “What does equals mean?” the room often goes quiet.

  The word “equals” can become circular and meaningless if we don’t anchor it to understanding. For many students, it’s just vocabulary without a mental model attached.

  So instead of saying:

  ❌“Two plus five equals seven,”

  Dr. Powell recommends saying:

  ✅“Two plus five is the same as seven.”

  That phrase—is the same as—carries meaning students already understand. Most children grasp the concept of “same” long before they encounter formal equations.

  This subtle language shift reinforces that the equal sign describes a relationship between two quantities.

  Expose Students to More Than One Format

  Dr. Powell also emphasizes that we must move beyond the typical equation format students see 80–90% of the time:

  2 + 5 = 7

  In this format, the equal sign sits in what she calls the “second-to-last position,” followed by the answer. If this is all students ever see, they naturally associate the equal sign with “write the answer here.”

  Instead, we should regularly expose students to:

  7 = 5 + 2

  And read it aloud as:

  ✅“Seven is the same as five plus two.”

  When students hear and see equations structured differently, they begin to internalize that the equal sign is not directional — it does not mean “the answer comes next.” It signals balance.

Understanding Equation Types: Standard vs. Non-Standard

To help parents and teachers recognize what builds relational thinking, Dr. Powell describes different types of equations.

1. Standard Equations

These are the most common:

Number + Number = Number

Examples:

2 + 5 = 7

14 × 2 = 28

They contain:

• One operator (plus, minus, multiply, divide)
• One equal sign
• Three positions for numbers

Students see these constantly. While important, they do not, on their own, build deep relational understanding.

2. Identity Statements
These are simple but powerful:

3 = 3

14 = 14

They’re often used with manipulatives or drawings to show equivalence visually. Though less common in worksheets, they reinforce the idea that both sides can represent the same quantity.

3. Three-Position Non-Standard Equations
These flip the standard format:

7 = 5 + 2

Here, the equal sign appears earlier in the equation. Students must interpret the statement relationally rather than procedurally.

4. Four-Position Non-Standard Equations (The Real Game-Changers)

This is where relational thinking truly develops:

2 + 3 = 4 + 1

Or:

2 + _ = 4 + 1

Now there are numbers on both sides of the equal sign.
Students cannot simply “do something.” They must compare both sides and think about balance.

For example:

4 + 1 = 5

So what makes the left side equal 5?

2 + _ = 5

The missing number is 3.

Many students with an operational mindset will instead add everything they see and write 7, revealing that they are still thinking procedurally rather than relationally.

Practical Takeaways for Parents and Teachers

To build relational understanding:
🔴Say “is the same as” instead of relying solely on “equals.”

🟠Read equations in both directions.

🟡Regularly include non-standard equation formats.

🟢Use manipulatives or drawings to show balance.

🔵Ask: “How do we know both sides are the same?”

When students begin to see the equal sign as a symbol of balance rather than a command to compute, their mathematical thinking shifts in powerful ways.

Starting at the True Beginning of “Equal”

As the conversation continues, Adrianne highlights something powerful: when she and Heather were writing their Math Facts to Memory book, focused on addition and subtraction within 10, they realized something surprising.

They couldn’t start with addition.

They had to start with equal.

When Facts Don’t Stick, Look Earlier

Heather shares that many of the students they teach, particularly those with dyslexia and dyscalculia, struggled to learn addition and subtraction facts not because they lacked effort, but because they lacked conceptual grounding.

If a child does not truly understand what the equal sign represents, then:

2 + 3 = 5

is just a string of symbols to memorize.

So instead of beginning with operators, their book begins with something simpler:

• Two sides
• An equal sign
• The concept of equal and unequal
• No addition or subtraction at all

 
For many students, this identity-level work, like:

3 = 3

It was not optional. It was foundational.

Dr. Sarah Powell affirms this insight immediately. Students need repeated experiences seeing that:

“These three bears are the same as these three bears.”

Too often, she explains, students aren’t explicitly taught the meaning of the equal sign. Instead, they are conditioned by repetition. If the first 100–200 times they see the equal sign they are asked to “add and write the answer,” we inadvertently train them into operational thinking.

We can condition relational thinking just as intentionally.

Moving from Manipulatives to Drawings to Abstraction

Adrianne then asks an important instructional question:

What does it look like to move students from hands-on manipulatives into more abstract representations?

Dr. Powell describes a clear progression used in her fourth-grade intervention work.

Step 1: Concrete Manipulatives

Students begin with:

• Balance scales
• Centimeter cubes
• Counting objects
• Number-line balance tools
  These physical tools make “balance” visible and tangible.

  Step 2: Drawing Representations

  Students quickly transition to drawing.

  Dr. Powell describes a simple setup:

• An equal sign in the middle of the page
• A large rectangle on each side

 
For an equation like: 4 + _ = 7

Students:

• Draw four circles on one side
• Draw seven squares on the other side
   
  Then they ask:

• Is 4 the same as 7? No.
• Add one circle. Is 5 the same as 7? No.
• Add another. Is 6 the same as 7? No.
• Add another. Now 7 is the same as 7.
   
  Now they can connect the drawing back to the equation:4 + 3 = 7

  4 + 3 is the same as 7.

  This repeated connection between:
  ✅Visual

  ✅Concrete

  ✅Symbolic

  ✅Verbal

  builds flexible understanding.

  And importantly, the abstract equation is always present alongside the drawing. Representations are never isolated.

  The Power of Saying It Out Loud

  Adrianne points out something subtle but critical:

  When students verbalize their reasoning — and echo back phrases like “is the same as” — they build the internal dialogue they’ll later need during independent work.

  Language becomes thought.

  For students with dyscalculia, who may struggle with working memory and internal processing, rehearsed mathematical language strengthens clarity and independence.

  Don’t Forget the “Equal Line”

  Toward the end of the discussion, Dr. Powell raises a point many educators overlook: the equal line in vertical computation.

  When students solve:
  25
  + 25
  ___
  50

  That horizontal line is functionally an equal sign.

  It communicates:
  25 + 25 is the same as 50.

  Since it often feels procedural, we rarely discuss its meaning.

  Dr. Powell suggests we should.

  Students should understand that:
  ✅The top expression and the bottom result represent equivalent quantities.

  ✅The “equal line” carries the same relational meaning as the horizontal equal sign.

  ✅And importantly, 50 is also the same as 25 + 25.

  For many learners, especially those with dyscalculia, the vertical format becomes their most common encounter with equivalence. If we never attach relational meaning to that line, we miss another opportunity to reinforce understanding.

Does the Equal Sign Really Impact Word Problems?

As the conversation wraps up, Adrianne circles back to something Dr. Sarah Powell mentioned earlier:

If understanding the equal sign is “just one small piece,” does it really make a difference in word problem solving?

The answer is nuanced and fascinating.

The Equal Sign Isn’t Everything — But It’s Not Nothing

Dr. Powell is clear:
Understanding the equal sign does not solve all word problem challenges.

Successful word problem instruction must include:
🟣A clear attack strategy

🔵Strong schema (problem type) instruction

🟢Practice identifying relevant information

Here’s where the equal sign enters the picture.

By third or fourth grade, most students represent word problems using equations. Even if they begin with graphic organizers, they quickly transition to writing equations to model what’s happening in the story.

If a student misunderstands what the equal sign represents, that misunderstanding can interfere with solving the equation correctly.

Check out our interview with Dr. Powell on math language and word problems here.

What the Research Showed

Dr. Powell describes a study conducted with third graders. Two groups received strong word problem instruction. The only difference?

One group received an additional five-minute daily component called Equation Quest, focused specifically on understanding the equal sign as relational.

During those five minutes, students:

• Used balance scales
• Drew representations
• Practiced varied equation types
• Replaced “equals” with “is the same as”
• Solved non-standard equations

The other group received a neutral filler activity instead.

Here’s what happened:
👉🏼Equal sign understanding improved
(Expected — they were explicitly taught it.)

👉🏼Equation solving improved
Students who practiced relational thinking became better at solving equations.

👉🏼Word problem performance improved
Those stronger equation skills led to better word problem outcomes.

Both groups made gains in word problem solving. But the Equation Quest group had a measurable advantage.

Not massive.
Not dramatic.
But consistent.

When researchers followed the students into fourth grade, that slight advantage remained.

Why This Matters Long-Term

Dr. Powell wonders aloud:
If students receive intensive relational practice in third grade, what happens later?
❓Does it support pre-algebra?

❓Does it make algebraic reasoning more intuitive?

❓Does it strengthen flexible equation solving in middle school?

While long-term data is still evolving, the logic is compelling.
Algebra is built on equivalence.
If students deeply understand that:

“This side is the same as that side,”

Then they are better positioned to reason through increasingly complex expressions.

A Nuanced but Meaningful Advantage

Dr. Powell emphasizes something important for parents and educators:

If you are not explicitly teaching the equal sign relationally, it is not catastrophic.

But when we:

• Pair schema instruction with relational equation work
• Include non-standard equation formats
• Reinforce “is the same as” language
• Connect concrete, visual, and symbolic models

We provide students with an additional layer of support.

For students with dyscalculia especially, these small conceptual reinforcements can create cumulative advantages over time.

As the episode closes, there’s laughter about how much depth exists inside “just one math symbol.”

But that’s the point.

Mathematical symbols are not decorative.

They carry meaning.

When that meaning is misunderstood, the ripple effects extend far beyond a single worksheet.

In this Unlocking Dyscalculia series, we keep returning to the same theme:

Struggles in math are often rooted in misunderstood concepts, not lack of intelligence or effort.

Sometimes, unlocking math doesn’t require a new program or harder problems.

It starts with redefining one small symbol:
The equal sign means “is the same as.”

You can connect with Dr. Sarah Powell by following her on social media @sarahpowellphd, or sending her an email at srpowell@utexas.edu.

Check out our other episodes of Unlocking Dyscalculia here:

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