But what about math?

Too often, math is treated as if it’s just one skill. People say things like, “I’m not a math person”—something you’d rarely hear someone say about reading. The truth is, being “good at math” is not an all-or-nothing talent you’re born with. Just like reading, math proficiency is made up of many connected skills that can be strengthened over time.

In this episode of Unlocking Dyscalculia, Adrianne Meldrum and Heather Brand dive into their vision for a Mathematical Proficiency Rope—a simpler, yet still deeply interconnected model that explains what it really takes to develop strong math skills.

Table of Contents

The Three Strands of Mathematical Proficiency

Adrianne and Heather break math proficiency into three key strands:

1️⃣Semantics – The big-picture concepts that make up number sense. This is about understanding how numbers work, how operations behave, and having a sense of magnitude—knowing whether a number is big or small, and how it fits in relation to other numbers.

2️⃣Procedural – The ability to accurately and efficiently use algorithms, solve problems, and communicate mathematical ideas. Some students may have strong number sense but struggle to explain their process or document their thinking in a way others can follow.

3️⃣Verbal – The often-overlooked language side of math. This includes math vocabulary, word problems, the ability to retrieve math facts, and even understanding the morphology of math terms. Heather notes that math fact recall is a verbal process in the brain, relying on the phonological loop—something most people don’t realize.

Strengths, Weaknesses, and the Balance of the Rope

Every student’s “math rope” looks different. Adrianne shares about her son, whose semantics strand is strong and thick, but whose procedural and verbal strands are thinner. He excels in computation and problem-solving, but struggles when language-heavy math tasks are introduced—like word problems or vocabulary-heavy concepts.

Heather emphasizes that this variation is normal: students can have strengths in one strand and weaknesses in another. The goal isn’t to make every strand identical, but to strengthen each one so the overall rope—the student’s math proficiency—becomes sturdy and resilient.

Strand One: Semantics – The “Know-How” of Numbers

In the Math Rope, the first strand—semantics—is all about understanding how numbers work. This is often called number sense, and it’s the foundation on which everything else in math is built.

When a student has strong number sense, they can:
➡️Recognize patterns and sequences.

➡️Understand place value (knowing the difference between 3 ones and 3 million).

➡️Estimate whether an answer makes sense.

➡️Reason through problems before ever touching an algorithm.

When number sense is weak, students may throw out answers without realizing they’re way off, or they might not see the connection between quantities and the symbols we use to represent them. This gap can make it hard to progress to more advanced skills—because the conceptual “why” behind math is missing.

When Weak Semantics Is a Red Flag

Persistent struggles in this area can sometimes point toward dyscalculia, especially if a student finds estimating, rounding, or reasoning about numbers unusually challenging. Other times, visual processing differences may be at play. In these cases, even helpful tools like manipulatives or visuals can become overwhelming if they’re cluttered or presented without careful structure.

Making Number Sense Click

Heather emphasizes that students—especially those with dyscalculia—need hands-on, visual learning to build an inner number line. This could mean:

➡️Using manipulatives (like cubes or counters) to make math concrete.

➡️Presenting visual models that are clean and clutter-free.

➡️Adjusting the orientation of number lines—some students grasp vertical number lines better than horizontal ones.

➡️Using subitized visuals (patterns like those on dice) to help students instantly recognize quantities without counting one-by-one. Learn more about subitizing here.

The key? Make visuals and manipulatives work for learning, not against it. Overly busy images or “cute” clip art can distract from the math concept itself.

The Power of Generic Manipulatives

Not all concrete tools are equally effective. For example, paper money is highly specific—it only connects to purchasing scenarios. If you want students to transfer their learning across contexts, it’s better to use generic manipulatives like cubes or counters that can stand for anything in a story problem.

Be Explicit, Not Just Exploratory

While discovery learning has its place, research shows that students with learning differences benefit most from explicit, guided instruction. Don’t just hand out manipulatives and hope the “aha” moment happens. Instead:

✅Tell students exactly what to do with them.

✅Ask targeted questions that connect the hands-on activity to the math idea.

✅Show the concrete and abstract side-by-side, so students map the physical experience to the numerical representation.

This is where CRA-I—Concrete, Representational, Abstract, Integrated—comes in. Just like in reading instruction, where graphemes are connected to their sounds, CRA-I connects the tangible with the symbolic in math. This integration helps ensure that abstract numbers hold real meaning for students.

Read More about the CRA-I Method Here

Strand Two: Procedural Fluency – Doing the Math Efficiently and Accurately

If semantics is the “know-how” of math, procedural fluency is the know-how-to-do-it. This strand is all about efficiently and accurately carrying out calculations, using algorithms, and communicating mathematical ideas on paper. It’s where the abstract side of math—numbers, symbols, and steps—comes into play.

When procedural fluency is strong, a student can:

👉Recall and apply algorithms without overloading working memory.

👉Follow the correct sequence of steps.

👉Keep track of place value and directionality.

👉Show their work in a way that’s clear and logical to others.

What It Looks Like When Students Struggle

Weaknesses here often surface in ways teachers might label “careless mistakes” or “lack of effort.” In reality, the problem often stems from deeper challenges, like:
➡️Working memory overload – losing track of where they are in a multi-step problem.

➡️Processing speed difficulties – taking so long on one part of a problem that accuracy slips elsewhere.

➡️Directionality issues – mixing up columns in multiplication or adding across rows instead of down.

➡️Dysgraphia – struggling to write numbers neatly or in the correct alignment, which disrupts calculation.

For some students, getting started at all can feel impossible—not because they’re lazy, but because remembering the order of steps is overwhelming.

Why Procedural Fluency Gets a Bad Rap

The term “procedural fluency” has sparked debate in education circles, but here’s what it really means in this context:

👉Efficient, accurate calculations that don’t drain working memory.

It’s not about teaching students to invent their own algorithms. While creative problem-solving has value, centuries of refinement have led to the algorithms we use today because they work consistently and prepare students for higher-level math. A “made-up” method might help in the short term but fall apart when applied to more complex problems like polynomial division.

It’s also not about relying solely on strategies like “doubles plus one” forever. These strategies help build early number sense, but when it comes to multi-digit operations, students need instant recall of math facts to avoid bottlenecks in working memory.

Tools to Support Sequencing and Memory

One powerful support is the use of “buttons”—small visual icons representing each step of an algorithm. Placed near a student’s non-writing hand, they can be physically touched in sequence to help track progress. For example, a division “button” set might include a division sign, multiplication sign, subtraction sign, and an arrow pointing down. This tactile, visual sequencing tool keeps students grounded in the process without overloading their memory.

Worked examples can also be useful, but for some students, scanning a full problem with all its steps can feel overwhelming. Buttons offer a quick-reference shortcut that students can mentally rehearse or even redraw from memory during a test. Over time, repeated use builds an internalized picture of the sequence so they can work independently.

Practice Matters—A Lot

It sounds obvious, but practice is a critical (and often overlooked) part of building procedural fluency. Teachers may move on too quickly after a student demonstrates initial understanding, leaving them stuck in the acquisition phase without enough repetition to reach true automaticity.

The solution? Build in opportunities for independent work—without prompts or cues—to check whether students can apply the process on their own. Follow this with quick, targeted feedback to correct errors before they become ingrained. This cycle not only cements the steps but also prepares students to use them fluidly in more complex math later on.

Building Fluency—Even When It’s Slow

For students with learning differences, mastering math facts and algorithms can be a slow process. And slow can mean years—not just months. But that doesn’t mean giving up or defaulting to calculators too soon. With consistent, explicit practice and the right supports, nearly every student can make meaningful progress in procedural fluency.

Supports That Make a Difference

To help students strengthen this strand of the math rope, educators can:

✅Provide visual cues and worked examples that clearly outline each step of an algorithm.

✅Teach mnemonics or acronyms (like ROMANS for solving story problems) to guide problem-solving.

✅Pair steps with gross motor actions—for example, a “karate move” for each part of the long division process—to make sequences more memorable.

✅Reinforce place value and directionality explicitly to prevent common errors.

Sometimes, it’s a quirky saying that sticks forever. Adrianne shares a story of a former student who, years later in college, still remembered her saying, “The Y-axis? It’s straight up, man!”—complete with the arm motion. Those little memory hooks can be surprisingly powerful for procedural recall.

Strand Three: Verbal Proficiency – The Language of Math

If procedural fluency is about doing the math, verbal proficiency is about talking and thinking in math. Math has its own language—specialized vocabulary, unique meanings for everyday words, and symbolic representations. For many students, especially those with dyslexia, auditory processing challenges, or slower processing speed, this can be a huge barrier.

Why Language Matters in Math

Words like equal, sum, product, and factor carry precise meanings that extend into higher math. Even small words such as is or of shift meaning in math contexts. Without direct instruction, students may misinterpret them—or skip over them entirely—because they seem unimportant in regular reading.

We can take vocabulary deeper by teaching morphology (roots, prefixes, and suffixes). For example:

Weaving the Strands Together – Building the Full Math Rope

Throughout this conversation, we’ve explored the three essential strands of mathematical proficiency:
1️⃣Semantics – the conceptual understanding of how numbers work.

2️⃣Procedural Fluency – the ability to efficiently and accurately use algorithms and represent work.

3️⃣Verbal Proficiency – the language skills that allow students to understand, express, and retrieve mathematical ideas.

When these strands are strong and intertwined, they form a sturdy “math rope”—a visual metaphor inspired by Scarborough’s Reading Rope. But if one strand frays or is missing altogether, students may struggle to access higher-level content, experience math anxiety, or feel like “math just isn’t for them.”

Unfortunately, many teacher preparation programs and math curriculums tend to emphasize one or two strands while overlooking the third. Some lean heavily into concrete, visual learning with manipulatives but neglect to explicitly teach algorithms. Others drill procedural fluency without connecting it to visuals or conceptual understanding. And the verbal strand—arguably the most underrepresented—often gets lumped into language arts, when in fact it’s essential to math success.

Why Balanced Instruction Matters

Just as in reading, missing skills in math can trigger the Matthew Effect—students who fall behind in one area get further behind over time because they can’t fully engage with new material. A student who can’t retrieve math facts efficiently may struggle to keep up in multi-step problem solving. A student who doesn’t understand math vocabulary may misinterpret word problems. And a student with conceptual gaps may rely on memorized procedures without understanding why they work, leaving them stuck when faced with unfamiliar problems.

The CRA Club was born out of our own “now what?” moments—those times when we finished a great training or read a fantastic math book but didn’t know exactly how to apply it with real students. CRA Club bridges that gap by showing exactly how to connect the concrete, procedural, and verbal components in the classroom or tutoring setting.

We’ve made it so you can spend your energy teaching rather than reinventing the wheel:
✅Ready-to-use vocabulary cards

✅Clear visuals for each step of algorithms

✅Examples of how to connect hands-on activities to abstract representations

The Takeaway

Every student has the potential to become mathematically proficient—not just “good at math,” but able to understand, perform, and communicate mathematics with confidence. It happens when we stop treating math as a single skill and instead nurture all three strands of the math rope.

Whether you’re a parent, teacher, or tutor, we invite you to explore how CRA Club can help you strengthen every strand for your students. Because when the rope is whole, students see what we’ve always known—they can do math.

Curriculum Alone Can’t Rescue

a Student Struggling With Math.

We have the tools and strategies

that help students with complex learning needs

finally understand it. Let us show you how.

MFM Authors