What Are Math Gaps? Why Your Child May Struggle With Math They Were Already Taught
Updated: Oct 11
By Susan Ardila, M.Ed. | October 2026
Your eighth grader is struggling with equations. Your high schooler can solve a rational equation when it looks exactly like the example from class, but change the problem just a little and suddenly everything falls apart.
So, naturally, it looks like an algebra problem.
But what if it isn't?
What if the real problem started years earlier, when fractions never completely made sense?
This is something I see frequently in my work with students who struggle in math: the place where a student is struggling today is not necessarily the place where the learning gap began. Sometimes the actual problem is hiding several layers underneath the math that finally made it visible.
So, what exactly are math gaps?
A math gap is a previously taught mathematical concept or skill that a student never fully mastered. It doesn't necessarily mean your child was never taught the material. Often, the student learned enough to complete the work at the time, but never developed a deep enough understanding to retain it, connect it to later concepts, or apply it when the math looked different.
And that last part is important.
Math is cumulative. Concepts don't politely disappear when the school year ends. They come back, usually wearing a slightly more complicated outfit.
A weakness that was fairly easy to work around in elementary school can become a much bigger obstacle in middle or high school as new mathematics begins depending on that earlier knowledge.
That's why simply tutoring a student on whatever they're struggling with right now doesn't always solve the problem. If a high school student is struggling with rational equations because fractions never truly made sense, doing 30 more rational equations may give them more practice, but it doesn't necessarily fix what's actually getting in their way.
Sometimes, before we can help a student move forward, we have to figure out where the math first stopped making sense.
This is the difference between simply helping a student through today's assignment and providing specialized math intervention designed to rebuild the foundations underneath the struggle.
Key Takeaways
A math gap is a previously taught concept or skill that never became stable, usable mathematical knowledge.
The place where a student is struggling today may not be where the gap began.
Students can have significant foundational math gaps even when they earn good grades.
Math gaps and dyscalculia are not the same thing.
Effective math intervention identifies the underlying concepts interfering with current learning instead of simply assigning more grade-level practice.
In This Guide
The Most Important Thing Parents Need to Understand About Math Gaps
Why Fractions Are Often the Gap Hiding Under an Algebra Problem
“My Child Is in Eighth Grade. Why Are You Teaching Fourth-Grade Math?”
Do Math Gaps Have to Reach 100% Mastery Before Moving Forward?
A Message to the Parent Who Is Watching Their Child Struggle
What Does a “Gap” in Math Actually Mean?
First, let's clear up one very common misunderstanding:
A math gap does not automatically mean, “My teacher never taught me this.”
Sometimes a student truly hasn't been taught a particular concept yet. If the skill is beyond what they've encountered in their curriculum, that's not a learning gap. It's simply something they haven't learned yet.
A foundational math gap, on the other hand, involves a concept or skill the student has previously encountered but never mastered well enough for that knowledge to become stable and usable.
There is an important difference between being exposed to a concept, being able to perform it temporarily, and actually understanding it.
A student may have successfully completed an entire unit by memorizing a procedure:
First I do this. Then I move this over here. Then I flip this. Then I multiply these. My teacher said that's what I'm supposed to do.
They may even get the correct answer.
But ask, “Why does that work?” and suddenly things get interesting.
Sometimes the answer is essentially, “Because that's what my teacher told me to do.”
That isn't necessarily mastery. It may simply mean the student became very good at following directions.
And to be fair to kids, forgetting something also doesn't mean they were never taught it.
As a former classroom teacher, I've actually had students tell me, “My teacher never taught us this last year.”
There was just one tiny problem with that argument:
I was their teacher last year.
I taught it. I remember teaching it. There may even have been colorful manipulatives involved.
What the student usually means is, “I don't remember learning this.” And that's actually useful information, because if the learning didn't stick, we need to figure out why.
This distinction becomes especially important for students who struggle in math. A student can appear to “know” a skill because they can reproduce a memorized procedure under familiar conditions. But if the understanding underneath that procedure is fragile, changing the wording, representation, unknown, or context may suddenly leave them unsure what to do.
True mathematical mastery involves more than getting the right answer. A student needs to understand a concept well enough to recognize it, use it, and apply it in different contexts.
When that underlying understanding never fully develops, that's where a seemingly small gap can quietly follow a student into much more advanced mathematics.
The Most Important Thing Parents Need to Understand About Math Gaps
The struggle you see today may not be where the problem started.
If there is one thing I wish more parents understood about math intervention, it is this:
We cannot assume that the math a child is struggling with right now is where we need to begin.
I want to find the first point where the math stopped making sense.
That's because mathematics is cumulative. Later concepts repeatedly draw upon earlier ones. The progression isn't perfectly linear for every child, but broadly speaking, mathematical understanding develops through interconnected concepts:
Number sense → operations → multiplication and division → fractions → ratios and proportions → equations → algebraic reasoning
If something underneath isn't secure, everything built on top of it becomes harder to access.
This is why I can have a student come to me because they're struggling with Algebra I, Algebra II, or another upper-level math course, and discover that the most important thing I can do for them initially has very little to do with their current chapter.
This is why a student can come to me struggling with Algebra I, Algebra II, or another upper-level math course and discover that the most important thing we need to address initially began years earlier.
Sometimes that missing foundation is fractions.
That can be difficult for a parent to hear. My child is in high school. Why are we working on elementary-school fractions when they need help with algebra?
Because fractions didn't disappear when elementary school ended. They followed the student straight into algebra.
We'll look more closely at that relationship below, but the important point is this: practicing the visible struggle does not necessarily address the source of the struggle.
This isn't just something I observe in my own students. In its 2008 final report, the National Mathematics Advisory Panel identified fluency with whole numbers, fluency with fractions, and particular aspects of geometry and measurement as critical foundations for algebra. The Panel's conclusions drew on multiple sources of evidence, including international curriculum comparisons and a national survey of 743 teachers of introductory algebra about what students need to know to succeed in algebra.
This is why effective math remediation sometimes means going further back than anyone expected.
Not because we're trying to make an eighth grader redo fourth grade.
Not because we're ignoring grade-level mathematics.
And definitely not because I have a secret desire to make teenagers spend the rest of their natural lives reviewing fractions.
We're looking for the specific foundational concepts that are preventing the student from accessing the mathematics they're expected to learn now.
I don't remediate based on the grade level printed on a child's report card. I remediate from the point where the math stopped making sense.
That may mean going back one unit, one year, or sometimes several years. But the goal isn't to stay there.
The goal is to repair the specific foundation preventing the student from moving forward.
If we continue adding increasingly complex mathematics on top of something the student never understood, eventually the structure starts to wobble.
At some point, we have to stop adding another floor and fix what's underneath it.
Can a Child Have Math Gaps and Still Get Good Grades?
Absolutely. A student can earn good grades in math and still have significant foundational gaps.
This is one of the reasons math gaps can remain hidden for years.
The Procedural Mask: When Good Grades Hide Weak Math Understanding
The procedural mask is what I call the appearance of mathematical mastery created when a student can successfully imitate procedures without understanding the mathematics underneath them.
Some students become incredibly good at wearing this procedural mask. On paper, everything looks fine. They're passing their classes. They may even be earning As and Bs. But underneath those grades, they may be relying heavily on memorized procedures rather than actually understanding the mathematics they're doing.
These students often become very good at figuring out the rules of the game.
They memorize a formula. They imitate the example their teacher just demonstrated. They remember an algorithm long enough to use it on the test. They recognize that this kind of worksheet means I follow these steps.
And sometimes that works surprisingly well.
Until it doesn't.
Change the problem just enough that the memorized procedure no longer fits perfectly, and suddenly the student doesn't know what to do.
That's why performance is not necessarily the same thing as mastery.
If a student gets 8 out of 10 problems correct, that's certainly useful information. But I also want to know how they got those eight problems correct.
Can the student explain why the strategy works?
Can they recognize when that strategy should be used without someone telling them?
Can they apply the same mathematical concept when the problem is presented differently?
Can they connect what they're learning now to something they already understand?
Those questions tell me much more about a student's mathematical understanding than a score by itself.
This distinction is also reflected in mathematics education research. The National Research Council's Adding It Up describes mathematical proficiency through five interconnected strands, including both conceptual understanding and procedural fluency. In other words, mathematical proficiency requires more than successfully executing a procedure.
Procedures aren't the enemy. Students need efficient procedures and algorithms, and I teach them all the time.
The problem occurs when the procedure is all the student has.
If I teach a child five steps without helping them understand the mathematics those steps represent, I've given them directions. I haven't necessarily given them knowledge they can use independently.
This is one reason parents sometimes tell me something that sounds completely contradictory:
“I don't understand. She knew how to do this last year.”
Or she may have known how to perform it under the conditions in which she learned it.
Those are not always the same thing.
When mathematical learning has meaning, students can begin making connections between concepts. They have something to reason from when they forget a step. They can adapt what they know when a problem changes.
When learning consists primarily of memorized procedures, there may be very little for the student to fall back on once that procedure is forgotten.
And that's how a student can bring home perfectly respectable grades while foundational gaps continue quietly growing underneath them.
What Does a Hidden Math Gap Look Like?
Hidden math gaps don't always announce themselves with a failing report card.
Sometimes they're much sneakier than that.
A student may complete homework, pass tests, and appear to be keeping up with the class while relying on strategies that conceal how fragile their understanding actually is.
In my sessions, I often uncover these gaps when I do something very simple:
I change the problem.
Not the underlying mathematics. Just the way the student has to think about it.
For example, suppose a student has learned a formula and can successfully plug in the numbers to find the answer. Great.
But then I give the student the answer and ask them to find one of the missing parts of the formula instead.
Same mathematical relationship. Same information. Different unknown.
And suddenly I hear:
“Wait...what do I do?”
That moment tells me something important.
The student may have learned how to follow the procedure, but they haven't necessarily developed enough understanding of the mathematical relationship to manipulate it independently.
There are several other patterns I frequently watch for:
A student can multiply accurately but struggles to explain what multiplication represents.
They know fraction procedures but have difficulty reasoning about fraction magnitude.
They perform a skill successfully during instruction but seem to have “forgotten everything” a few weeks later.
They solve familiar problems correctly but struggle when the same concept appears in a word problem, visual model, number line, or more complex application.
They rely on superficial clues such as “Does this word mean I multiply?”, “Is this where I flip the fraction?”, or “Do I move the decimal here?” rather than reasoning about what is happening mathematically.
None of these behaviors automatically proves that a particular math gap exists. But they are reasons to investigate beneath surface-level performance.
One of the biggest warning signs is this:
The student knows what to do only when the problem looks the way it looked when they learned it.
That's a problem because mathematics doesn't stay neatly separated into worksheet-sized boxes.
As students move into more advanced mathematics, concepts begin interacting. Fractions show up inside equations. Ratios connect to proportions. Proportional reasoning connects to slope. Operations that were once taught individually become tiny pieces of much larger problems.
Eventually, no worksheet heading is going to announce:
“GOOD NEWS! THIS IS A FRACTION PROBLEM. PLEASE USE THE FRACTION PROCEDURE YOU MEMORIZED IN FIFTH GRADE.”
The student has to recognize the mathematics for themselves.
That's why I pay close attention to transfer and generalization when I'm determining whether a skill has actually been mastered.
Can the student take what they've learned and use it somewhere else?
If the answer is consistently no, then even if they're getting the familiar problems correct, I start looking for what understanding may still be missing underneath the procedure.
Because sometimes the most important math gaps aren't the ones producing wrong answers.
They're the ones a student has become very good at hiding.
The Math Gap Map: Where You See the Struggle vs. Where the Gap May Begin
The skill causing the most visible difficulty isn't always the skill that needs to be remediated first.
What the student is struggling with now | Foundational concepts I may investigate |
Rational equations | Equation solving → fraction operations → fraction magnitude |
Slope | Ratios and proportions → fractions → multiplication/division relationships |
Factoring | Factors and multiples → multiplication/division relationships |
Fraction operations | Fraction magnitude → equivalence → multiplication/division concepts |
Multi-step word problems | Operations → mathematical language → representation → reasoning |
This is not a diagnostic chart or a universal learning sequence. Two students can struggle with the same grade-level concept for completely different reasons. The purpose of the Math Gap Map is to illustrate why effective intervention sometimes needs to look beneath the student's current coursework to determine where understanding actually began to break down.
Why Math Gaps Tend to Grow Instead of Disappear
One of the most frustrating things about math gaps is that they rarely stay neatly contained.
A child doesn't usually struggle with one isolated concept in fourth grade, leave it behind, and continue happily through mathematics as though nothing happened.
Math has a rather inconvenient habit of bringing old concepts back.
Each year, students aren't simply learning a brand-new collection of unrelated skills. Many of the concepts they've already encountered return in increasingly sophisticated forms. The mathematics becomes more abstract, more interconnected, and usually adds another layer of complexity.
So if the earlier version of a concept never made sense, adding complexity doesn't repair it.
It usually makes the gap more visible.
Think about how mathematical understanding can develop over time:
Equal groups → multiplication → fractions and ratios → proportional reasoning → algebra
Or consider fractions:
Fraction magnitude → fraction operations → proportions → slope → rational expressions and equations
These aren't meant to suggest that every child learns mathematics in one perfectly straight line. They don't. But later mathematics repeatedly calls upon knowledge students were expected to develop earlier.
That's why I sometimes hear from parents who are genuinely confused:
“She did fine in elementary school. I don't understand why math suddenly became so hard.”
Usually, it didn't suddenly become hard.
What changed may be the student's ability to compensate for what they didn't understand.
In earlier grades, a student may work around weak understanding by counting on their fingers, memorizing enough facts to pass a unit, copying the teacher's procedure, or using worksheet cues to determine which operation to perform.
Middle and high school mathematics are less accommodating.
Several previously learned skills can now appear inside one problem. A student working with an equation may simultaneously need fractions, negative numbers, order of operations, and algebraic reasoning.
A foundational weakness that was once possible to work around can suddenly become impossible to hide.
The gap didn't necessarily appear overnight. The mathematics finally became complex enough that the student could no longer work around it.
Students certainly continue developing, and some misunderstandings resolve with appropriate instruction and experience. But repeatedly adding more complex mathematics does not automatically repair an unresolved foundation.
Eventually, we need to identify what's missing and teach it.
Not every standard from every previous grade.
Not years of worksheets.
The specific foundational mathematics preventing the student from accessing what comes next..
Why Fractions Are Often the Gap Hiding Under an Algebra Problem
If I could choose one area of elementary mathematics that I would never want a student to simply memorize their way through, fractions would be very high on that list.
Fractions have a way of showing up again.
And again.
And then, just when you thought they had finally gone away, algebra invites them back with letters attached.
I've worked with multiple students who came to me because they were struggling with Algebra I, Algebra II, or other upper-level mathematics. Naturally, the assumption was that they needed help with their current grade-level material.
And they did.
But once I began looking underneath the immediate struggle, I found something else:
Their fraction understanding had never been solidified.
This is exactly why identifying the source of a math gap matters.
If a student is struggling to solve an equation containing fractions, I can certainly teach the procedure for solving that particular type of equation. With enough prompting and repetition, the student may even become pretty good at it.
But if fractions themselves still don't make sense, I'm asking that student to reason algebraically with numbers they don't fully understand.
That's a much bigger problem.
A student with strong fraction understanding doesn't just know a collection of rules such as find a common denominator, multiply straight across, or invert and multiply.
They understand that fractions are numbers with magnitude.
They understand relationships between fractions.
They can reason about whether an answer makes sense.
They recognize equivalent quantities.
They understand what happens to a quantity when it is multiplied or divided by a fraction.
Those ideas matter far beyond the elementary fraction unit.
Research supports that connection. In its 2008 final report, the National Mathematics Advisory Panel identified fluency with fractions as one of the critical foundations for algebra. In a separate longitudinal study, Siegler and colleagues analyzed data from more than 4,000 students and found that earlier knowledge of fractions and division uniquely predicted later algebra knowledge and overall mathematics achievement, even after controlling for earlier mathematical knowledge and several other factors.
That distinction between knowing the procedure and understanding why it works is especially important for the students I work with.
Take the classic fraction-division rule:
Keep, change, flip.
Many students can chant it beautifully.
But ask them what they're actually doing or why dividing by a fraction involves multiplying by its reciprocal, and you may get the mathematical equivalent of crickets.
The mnemonic isn't inherently bad. It can be a useful memory aid after the mathematics makes sense.
The problem is when the mnemonic becomes the student's entire understanding of fraction division.
Because later, when fractions are embedded inside algebraic expressions, proportions, functions, rational equations, or other multi-step problems, there isn't always a cute little rhyme available to rescue them.
This is why I sometimes work on what appears to be “elementary math” with a middle or high school student.
We're not going backward for the sake of going backward.
We're fixing something that is actively interfering with the mathematics they're trying to do now.
And when that underlying fraction gap begins to close, something interesting often happens: the grade-level mathematics that seemed impossibly complicated starts becoming more accessible.
The algebra didn't suddenly get easier.
The student finally had more of the mathematics they needed in order to understand it.
Are Math Gaps the Same Thing as Dyscalculia?
No. Having gaps in math does not automatically mean a child has dyscalculia.
I want to make that distinction very clear because struggling in math and having dyscalculia are not interchangeable.
Math gaps describe missing or unstable mathematical learning. Dyscalculia is a specific learning difficulty affecting the acquisition of mathematical skills. A child can have math gaps without dyscalculia, and a student with dyscalculia can have math gaps as part of a much broader learning profile.
Any student can develop a math gap.
A child may have missed important instruction because of an extended absence. A class may have moved through a concept before the student was ready. The student may have understood most of a unit but missed one critical piece. Sometimes a relatively isolated concept simply never clicked.
In those situations, the gap may be fairly specific. Once we identify what is missing, explicitly teach it, and give the student appropriate opportunities to practice and apply it, the student may pick it up relatively quickly.
Dyscalculia can look very different.
In my work with students with dyscalculia, I frequently see gaps that aren't confined to one recent unit or even one grade level. When I begin looking underneath the student's current difficulties, the gaps may extend much further back into foundational concepts involving number relationships, quantity, place value, operations, or other early numerical concepts.
And importantly, dyscalculia does not look exactly the same in every child.
In their 2020 review, Learning Disabilities: Developmental Dyscalculia, researchers Elisa Castaldi, Manuela Piazza, and Teresa Iuculano describe developmental dyscalculia as a heterogeneous, multicomponent learning disability. In other words, there isn't one single dyscalculia profile. Difficulties may involve basic numerical concepts and operations, while broader cognitive processes that support mathematics, including attention, memory, and cognitive control, may also contribute to an individual student's profile.
That matters because you will sometimes see dyscalculia reduced online to something like:
“Dyscalculia is a number-sense deficit.”
That's far too simplistic.
Number sense can absolutely be an area of significant difficulty for a student with dyscalculia, and it's something I pay very close attention to. But dyscalculia is more complex than one deficit that presents identically in every student.
This is also why two children can both be “behind in math” and need very different intervention.
One student may have a relatively isolated gap from missed or incomplete instruction. Teach the missing concept in a clear way and suddenly they're off and running.
A student with dyscalculia may require much more individualized instruction, multiple representations of the same concept, smaller instructional steps, significantly more opportunities to connect ideas, and ongoing review before that learning becomes stable.
From the outside, both students may appear to have the same problem:
They're behind in math.
But “behind” tells me where the student is performing. It doesn't tell me why they're there.
And the why matters enormously when deciding what kind of instruction that child actually needs.
If you're wondering whether your child's persistent math difficulties could be related to dyscalculia, you can also use my free dyscalculia screener as a starting point.
If your child has already received a dyscalculia diagnosis and you're wondering what to do next, I've also put together a detailed parent guide explaining how to understand the evaluation, explore appropriate supports, and begin building an effective intervention plan.
Why “Just Practice It More” Often Doesn't Fix the Problem
If a child doesn't understand something in math, one of the most common solutions is also one of the most logical:
Give them more practice.
Sometimes that's exactly what they need.
But sometimes it isn't.
Imagine I hand you a passage written entirely in a language you don't understand.
You read it.
Nothing.
So I hand it back and tell you to read it again.
Still nothing.
Apparently, the obvious solution is another worksheet. So now you get to read it 20 more times.
Congratulations. You are now extremely practiced at not understanding that language.
The problem was never that you hadn't looked at the passage enough times. The problem was that you needed someone to help you make sense of it.
Math can work the same way.
If a student didn't understand the mathematical concept the first time, simply repeating the same explanation, procedure, or practice problems may give them more exposure without giving them more understanding.
This is especially important when I work with students with dyscalculia.
If a strategy didn't make sense to a student's brain the first time, my solution isn't automatically to explain the exact same thing louder, slower, and six more times.
Something has to change.
Maybe I bring out manipulatives so the student can physically build what the numbers represent.
Maybe we move from concrete objects to visual representations before asking the student to work with abstract symbols.
Maybe I break one lesson into much smaller instructional pieces.
Maybe the strategy that works beautifully for most students simply isn't the strategy that is going to work for this particular child.
So I try another one.
And another one if I need to.
That's part of specialized math intervention: having a large enough instructional toolbox that when one path doesn't reach a student, you have somewhere else to go.
This is one reason I incorporate multisensory math instruction into my intervention approach. Using concrete materials, visual representations, mathematical language, and explicit instruction gives students different ways to develop an understanding of concepts that previously didn't make sense.
And there is research behind using these kinds of approaches with students who struggle significantly in mathematics.
In a 2019 systematic review, Lafay and colleagues examined 38 studies involving manipulative-based interventions for students with mathematics learning disabilities. All 38 reported immediate improvement statistically or descriptively. However, only five studies were designed to isolate the specific added value of manipulatives, and the authors emphasized substantial methodological limitations in the evidence base.
In other words, this research does not tell us that handing a child manipulatives magically fixes a math learning disability.
A separate 2020 meta-analysis by Peltier and colleagues examined 53 single-case studies involving mathematics manipulatives with students who were at risk or identified with disabilities. Across those studies, the researchers reported a large overall effect, with a Tau-U effect size of 0.91 and a 95% confidence interval of 0.87 to 0.95.
But here's the part I think is especially important for parents:
The manipulative itself isn't the intervention.
I can put fraction tiles in front of a child all day long. If I don't use them purposefully to help that student connect the physical model to the visual representation, the mathematical language, and eventually the abstract numbers and symbols, I've basically just given them colorful plastic.
The goal is to make the mathematics visible and meaningful.
That's why my instruction often moves between concrete manipulatives, visual representations, explicit explanations, guided practice, different strategies, and eventually abstract mathematics. I also continuously spiral previously learned skills back into later sessions rather than assuming that because we learned something on Tuesday, it has now been permanently filed away in the brain under Math: Solved Forever.
Wouldn't that be convenient?
For students with dyscalculia, learning often requires more intentional connections, more opportunities to revisit concepts, and instruction designed around how that particular student understands mathematics.
That's very different from simply assigning more problems.
Practice strengthens learning that is beginning to make sense. Practice cannot, by itself, create understanding that was never there.
Sometimes the student doesn't need more of the same.
They need someone to figure out what isn't making sense, why it isn't making sense, and another way to make the mathematics accessible.
How Do I Find Out Where My Child's Math Gaps Actually Are?
If a student is struggling in math, the first question shouldn't simply be:
What are they getting wrong?
I want to know:
Where did the understanding begin to break down?
And honestly, I don't believe in spending six weeks playing mathematical detective if I have a tool that can give me a much more efficient place to start.
I start with an adaptive math diagnostic.
An adaptive diagnostic adjusts the difficulty of questions based on a student's responses. When a student demonstrates strength in an area, the assessment can move into more advanced material. When the student begins struggling, it can dig further into prerequisite skills.
For a student with potential gaps spanning several grade levels, that's extremely useful.
If a seventh grader is struggling with grade-level math, I don't want to assess only seventh-grade standards. That tells me what the student cannot currently do, but it may tell me very little about why they cannot do it.
And I already know they're struggling with seventh-grade math. That's probably why their parent called me in the first place.
I need to know what is happening underneath it.
An adaptive diagnostic gives me a much more efficient starting point by identifying patterns of relative strength and weakness across mathematical domains and levels. Instead of randomly going backward until something finally clicks, I have data pointing me toward the areas that deserve a closer look.
But here's the important part:
An assessment is a starting point, not the entire answer.
A diagnostic can tell me that a student performed poorly in a particular area. It cannot always tell me why.
Two students can miss the exact same fraction problem for completely different reasons.
One student may not understand fraction magnitude.
Another may understand fractions perfectly well but make a multiplication error.
Another may know exactly what to do but lose track of a step because of working-memory demands.
Another may have memorized a procedure but have no idea what the procedure actually represents.
Same wrong answer.
Very different instructional needs.
That's where professional interpretation becomes important.
Diagnostic data tells us where to investigate. Student thinking tells us what is actually happening.
Once I begin working with a student, I'm paying attention to far more than whether an answer is right or wrong.
I watch the strategies they choose. I listen to how they explain their thinking. I look for misconceptions. I notice whether they can select a strategy independently or wait for me to tell them what to do.
I change representations.
Can they understand the concept with manipulatives but not with numbers and symbols? Can they solve it visually but not verbally? Can they explain it to me but struggle to record the mathematics?
I change the context.
Can they use the skill only when the problem looks familiar, or can they recognize the same mathematical relationship when it appears somewhere else?
And I change my instruction.
If one explanation doesn't work, what happens when I approach the concept differently?
All of that gives me information that a score alone cannot provide.
This is particularly important when evaluating students with persistent mathematics difficulties. The Institute of Education Sciences' 2021 practice guide, Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades, developed from a review of the intervention research, recommends using assessment data to identify students needing intervention and regularly using student performance data to determine whether instruction should be adjusted. The guide also emphasizes systematic instruction, mathematical language, concrete and semi-concrete representations, number lines, and deliberate practice as components of effective mathematics intervention. IES Practice Guide: Assisting Students Struggling with Mathematics
That's also why I don't use a diagnostic score as a label for a child.
A percentile doesn't teach me how a student's brain makes sense of mathematics.
Assessment narrows the search. Working with the student reveals the story.
And once I understand that story, I can make much better decisions about where intervention actually needs to begin.
“My Child Is in Eighth Grade. Why Are You Teaching Fourth-Grade Math?”
I understand why this question makes parents uncomfortable.
You hired someone because your eighth grader is struggling with eighth-grade math. Then suddenly you're watching your child work on a concept they supposedly learned four years ago.
It can feel like we're going backward when everyone desperately wants the child to catch up.
But sometimes, going back to the right place is the fastest way forward.
My approach is simple:
I don't remediate based on the grade level printed on a child's report card. I remediate from the point where the math stopped making sense.
That does not mean I take an eighth grader, open a fourth-grade curriculum, start on page one, and announce that we'll see everyone again sometime around graduation.
That wouldn't be targeted intervention. That would be a very long field trip.
The goal is to identify the specific foundational concepts interfering with the student's ability to access current mathematics and remediate those concepts strategically.
Suppose an eighth grader is struggling with equations, proportional relationships, slope, and other grade-level concepts, and assessment reveals that their fraction understanding was never solidified.
I can ignore the fraction gap because fractions are considered “elementary math” and continue helping the student through each new eighth-grade assignment.
Or I can address the mathematical weakness that keeps showing up inside those assignments.
I'm choosing the second one.
Addressing fractions isn't taking that eighth grader backward. It's giving them access to mathematics that has been inaccessible because the foundation was never secure.
And there is strong longitudinal research supporting the importance of those earlier skills.
In a 2012 study published in Psychological Science, Robert Siegler and colleagues analyzed large, nationally representative longitudinal datasets from the United States and United Kingdom. Students' knowledge of fractions and division at age 10 predicted their algebra knowledge and overall mathematics achievement at age 16, even after the researchers statistically controlled for factors including intelligence, working memory, family income and education, and earlier mathematics achievement.
That's a six-year connection between earlier mathematical knowledge and later achievement.
So when I tell a parent that we need to address fractions before their child can become more independent with algebra, I'm not abandoning algebra.
I'm addressing one of the things preventing the student from accessing it.
This doesn't mean students must achieve perfect mastery of every previous skill before they're allowed to touch grade-level mathematics again.
We identify the highest-priority gaps, strengthen them, connect them directly to current mathematics, and continue moving forward while deliberately revisiting earlier learning.
Sometimes we're working below grade level.
Sometimes we're working at grade level.
Often, we're intentionally connecting the two.
The goal isn't to make an eighth grader do fourth-grade math. The goal is to make sure a fourth-grade gap doesn't follow that student into ninth grade, tenth grade, Algebra II, and beyond.
Do Math Gaps Have to Reach 100% Mastery Before Moving Forward?
Not necessarily.
When I'm remediating a foundational math gap, my goal is not to hold a student on one skill until they perform it perfectly every single time.
In my own practice, I generally begin feeling comfortable moving forward when a student is demonstrating roughly 70–80% mastery, provided I also see enough conceptual understanding to support what we're going to learn next.
I want to be very clear about that number, though.
Seventy to eighty percent is not a universal research-based mastery cutoff, diagnostic standard, or magical mathematical threshold. It is an instructional decision-making guideline I use when determining when my students are ready to begin moving forward.
And I don't base that decision on accuracy alone.
I also want to know: Does the student understand what they're doing? Can they explain at least some of their reasoning? Are they becoming more independent? Can they use the skill outside the exact context in which I taught it? Are their remaining mistakes signs of a major conceptual misunderstanding, or are we dealing with smaller errors that continued practice can reasonably address?
That distinction matters.
If a student gets 80% correct but clearly doesn't understand the mathematics, I'm not moving on simply because we've hit my preferred percentage.
Conversely, if a student demonstrates strong conceptual understanding but makes an occasional computational mistake, I'm probably not going to keep them parked on the same concept for three additional weeks in pursuit of a beautiful 100%.
Why?
Because moving forward doesn't mean we stop working on what came before.
One of the most important parts of my intervention approach is intentional spiral review.
Skills don't disappear from our sessions because we've officially “moved on.” I deliberately bring previously learned concepts back over time, sometimes independently, sometimes mixed into new learning, and sometimes in a different representation or context.
That allows us to continue strengthening the remaining learning while still giving the student access to new mathematics.
There is a strong learning-science rationale for revisiting material over time rather than relying only on concentrated practice. In a major 2013 review, John Dunlosky and colleagues evaluated ten commonly used learning techniques and rated distributed practice and practice testing as high-utility approaches because their benefits generalized across learners, materials, and educational contexts.
A 2006 meta-analysis by Cepeda and colleagues, synthesizing 839 assessments across 317 experiments, likewise found a robust advantage for distributed over massed practice in verbal recall. This research doesn't prescribe a particular mathematics-intervention schedule, but it supports the broader learning principle behind intentional spiral review: learning benefits from being revisited over time rather than practiced once and abandoned.
That's essentially the balance I'm trying to create:
Forward movement + planned revisiting.
If I wait for absolute perfection before introducing anything new, a student who is already behind can spend an enormous amount of instructional time trying to squeeze the final few percentage points out of one skill.
But if I introduce a concept on Monday, declare it “mastered” on Friday, and never mention it again, fragile learning can disappear surprisingly quickly.
Neither extreme makes much sense to me.
I want the student's understanding secure enough to support what comes next, and then I want to keep strengthening that knowledge as we move forward.
Because remediation shouldn't trap a student in the past.
The entire purpose of repairing foundational gaps is to help the student move forward.
The Damage Isn't Only Academic
Math gaps don't just affect math.
After enough years of struggling, they can begin changing the way a child sees themselves as a learner.
I see this frequently with students who come to me after experiencing repeated difficulty in mathematics.
I put a problem in front of them.
Before they've really looked at it, I hear:
“I don't know.”
Sometimes I know they do know.
I've watched them solve something similar. We've discussed the concept. They have enough knowledge to at least begin.
But “I don't know” has become automatic.
Why?
Because after being wrong enough times, not trying can start to feel safer than trying and being wrong again.
If I say I don't know, I can't fail.
If I don't attempt the problem, nobody can see me get it wrong.
For some students, that response has been built through years of frustration.
The pattern can start looking something like this:
Repeated confusion → repeated failure → reduced confidence → avoidance → less participation → fewer opportunities to learn → even larger gaps
And now we're no longer dealing exclusively with mathematics.
We're dealing with a child who may have started developing beliefs about what their struggles supposedly say about them.
I've had students reach a point where the thinking underneath their behavior seems to be:
If I can't even do basic math, maybe I'm just stupid.
That breaks my heart, because a learning difficulty has somehow turned into a judgment about intelligence.
And it isn't one.
Research supports an association between mathematics anxiety and mathematics performance, although the relationship is more complicated than saying one simply causes the other.
In a 2019 meta-analysis published in Frontiers in Psychology, Zhang, Zhao, and Kong synthesized 84 independent samples representing 8,680 participants. They found a significant negative association between mathematics anxiety and mathematics performance (r = −0.28), meaning higher math anxiety was associated, on average, with lower mathematics performance.
Importantly, that doesn't mean every child with a math gap will develop math anxiety, or that poor performance inevitably causes anxiety.
Children are far more complicated than one tidy arrow on a research diagram.
But it does reinforce something I see constantly in practice: the academic and emotional sides of persistent math difficulty can become deeply intertwined.
That's why one of my earliest goals with a new student often has very little to do with a percentage on a worksheet.
I want them to discover two things.
First, making a mistake is not a catastrophe.
I make mistakes. Constantly.
Sometimes I'll make one during a session, catch it, fix it, and move on. I actually think there's value in students seeing that. Being good at math doesn't mean your pencil has been granted immunity from incorrect answers.
Mistakes give us information.
We figure out what happened, correct the misunderstanding, and keep going.
Second, I want the student to experience something they may not have experienced in math for a very long time:
Success that actually makes sense to them.
Not success because I walked them through every step.
Not success because they memorized what I said.
Success because something finally clicked.
And when that starts happening, the change can be surprisingly fast.
Parents often notice the confidence shift before they see a dramatic change in grades.
Their child dreads math less. Homework becomes less of a battle. They volunteer answers instead of immediately saying “I don't know.” They want to show someone what they learned. They start taking chances.
That willingness to try matters because every attempt, explanation, strategy, mistake, and correction creates another opportunity to learn.
The moment I get excited about isn't necessarily when a student starts getting everything right.
Sometimes it's much smaller.
I put a problem in front of them.
They pause.
They think.
And instead of automatically saying “I don't know,” they try.
What Does Successful Math Remediation Look Like?
Successful math remediation doesn't always announce itself with an A on the next test.
Of course, improved academic performance matters. We're not doing all of this just so everyone can have warm, fuzzy feelings about fractions.
But grades are often not the first thing I look at when I'm deciding whether intervention is working.
Some of the earliest signs are easy to overlook.
A student attempts a problem before saying “I don't know.” They explain their thinking. They catch a mistake. They ask a mathematical question. When one strategy doesn't work, they try another one.
Sometimes a parent notices something even more practical:
“We're not fighting over math homework anymore.”
Those changes matter. They suggest the student is beginning to interact with mathematics differently.
But I also want to be very clear about something:
Confidence isn't a substitute for mathematical skill. But a student who is finally willing to engage with mathematics has far more opportunities to develop it.
Telling a child “You can do it!” while continuing to give them mathematics they don't understand isn't intervention.
Confidence needs evidence.
Students need opportunities to experience genuine mathematical success: understanding a concept that previously confused them, solving something with less help, selecting a strategy independently, or realizing that they can make a mistake and still figure out what to do next.
That's how I want confidence to develop.
Not from empty reassurance.
From competence.
Over time, I also expect to see changes in the mathematics itself.
The student should begin requiring less prompting.
Their strategies should become more efficient.
Previously learned concepts should become easier to retrieve.
They should be able to use knowledge in contexts that look different from the original instruction.
And one of the biggest indicators for me is transfer.
If I teach a student a concept in one format and they can recognize and apply that same mathematical idea somewhere else, that's very different from memorizing one procedure for one type of problem.
The National Research Council's 2001 report Adding It Up describes mathematical proficiency through five interconnected strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition.
That final strand is particularly relevant here. Productive disposition involves seeing mathematics as sensible and worthwhile while believing that effort in mathematics can be effective.
That's very different from a student approaching every problem thinking:
There's no point. I'm going to get it wrong anyway.
So yes, I want grades and assessment scores to improve. I want greater accuracy and independence.
But I also watch what happens before those numbers move.
Does the student engage? Reason? Recover from mistakes? Recognize something learned weeks earlier and use it without being handed the strategy?
Those changes tell me we're not simply teaching the student how to survive the next assignment.
We're changing the way they understand and interact with mathematics.
And that's what gives improved academic performance a chance to last.
So, Can Math Gaps Actually Be Closed?
Yes. Math gaps can be remediated, but identifying the right gap matters.
More grade-level practice isn't necessarily the answer.
More worksheets aren't necessarily the answer.
And going back one chapter isn't necessarily far enough.
If a student's current difficulty is actually being caused by mathematics they never understood several years earlier, we can practice today's problem until both of us would happily launch the worksheet into the sun, but we still haven't addressed the underlying issue.
The question I keep coming back to throughout intervention is:
Where did the mathematics first stop making sense?
That's where I want to investigate.
Maybe the student struggling with Algebra II needs stronger equation-solving skills.
Maybe they understand equations but fractions are interfering.
Maybe the fraction difficulty is actually rooted in weak multiplication and division concepts.
And for some students with dyscalculia, we may need to go even further back and strengthen foundational understanding of numbers, quantities, and relationships.
The point isn't to automatically send every struggling student back to kindergarten.
The point is to go back as far as necessary, but no farther than necessary, to find the mathematics preventing that student from moving forward.
Then we teach it differently. We make the mathematics concrete or visual when needed, break complex ideas into smaller explicit pieces, connect new learning to what the student already understands, try another strategy when the first doesn't click, and deliberately revisit learning over time.
Most importantly, we help the student understand what they're doing and why they're doing it.
Research supports several instructional components commonly used in intensive mathematics intervention. In a 2009 meta-analysis published in Review of Educational Research, Gersten and colleagues synthesized 42 mathematics interventions for students with learning disabilities. Explicit instruction produced a large mean effect (g = 1.22), teaching students to use problem-solving heuristics produced a large mean effect (g = 1.56), and interventions incorporating visual representations showed a positive mean effect when visual approaches were combined (g = 0.47).
That doesn't mean every gap disappears quickly.
It doesn't mean every student will progress at the same rate.
And it certainly doesn't mean dyscalculia can be “cured” by finding the perfect worksheet, manipulative, or teaching strategy.
For some students, particularly those with dyscalculia and other learning differences, mathematics may continue to require specialized instruction, accommodations, explicit strategies, and more intentional review.
But persistent difficulty does not mean a student is incapable of mathematical growth.
I've watched students begin intervention convinced they simply “can't do math.”
Then something makes sense.
Then something else makes sense.
They begin making connections. They need less help. They start trying before asking what to do. Mathematics that once seemed inaccessible becomes something they can reason through.
That's why I don't view remediation as dragging a student backward.
We're going back to retrieve something they needed all along, then bringing it forward with us.
The goal isn't simply to fill a hole on an assessment report. It's to give the student enough understanding underneath their mathematics that they can finally keep moving forward.
A Message to the Parent Who Is Watching Their Child Struggle
If you've been watching your child struggle in math, it can be incredibly difficult to see them working on concepts you thought they learned years ago.
Your high schooler working on fractions isn't “going backward.”
Your seventh grader using manipulatives isn't “too old” for them.
And discovering that your child's current math difficulties trace back several grade levels doesn't mean they've somehow failed years of mathematics.
It means we've learned something important about where they need support.
Sometimes moving forward in mathematics requires finding the point where understanding broke down and repairing it.
That may mean temporarily stepping away from the chapter causing the most obvious frustration and asking a different question:
What does my child need to understand in order for this mathematics to make sense?
The goal isn't to make a student dependent on someone who can walk them through every assignment. It's to build enough mathematical understanding that they increasingly know what to do without someone standing beside them.
Sometimes that means slowing down before we can speed up. It means looking beyond today's incorrect answer and figuring out why today's mathematics doesn't make sense in the first place.
The goal isn't to reteach years of mathematics. It's to identify the foundational concepts standing between the student and the mathematics they're trying to learn now.
That's the work I do through MindBridge Math Mastery, where I provide individualized, specialized math intervention for students with persistent mathematics difficulties, including students with dyscalculia, ADHD, and other learning differences.
Rather than beginning and ending with a student's grade level, I use diagnostic data together with careful observation of the student's mathematical thinking to determine where understanding begins to break down and what needs to be strengthened.
For families who aren't sure where their child's difficulties are coming from, an adaptive math diagnostic assessment can provide a useful starting point for identifying strengths, potential foundational gaps, and areas that deserve closer investigation.
Because before we decide how to help a student move forward, we need to understand what has been holding them back.
Not Sure What's Really Behind Your Child's Math Struggles?
If your child is struggling in math and you're not sure whether you're seeing a few missing skills, deeper foundational gaps, dyscalculia, or a combination of challenges, you don't have to figure it out on your own.
Schedule a free strategy session with MindBridge Math Mastery.
We'll talk about what you're seeing at home and at school, your child's history with math, what has and hasn't worked so far, and what may be the most appropriate next step. If more information is needed, I can also explain whether an adaptive math diagnostic assessment would be useful.
The goal is to understand what's happening, answer your questions, and determine what type of support, if any, would be appropriate for your child.
Frequently Asked Questions About Math Gaps
What is a learning gap in math?
A learning gap in math is a previously taught mathematical concept or skill that a student has not fully mastered. The student may have been exposed to the material and may even have performed it successfully at one time, but the understanding isn't secure enough to retain, apply independently, or use in new contexts.
How can I tell if my child has gaps in math?
Possible signs include repeatedly forgetting previously learned skills, relying heavily on memorized procedures, struggling when a familiar problem is presented differently, difficulty explaining why a strategy works, or needing substantial help with skills taught in earlier grades. An adaptive diagnostic combined with observation of the student's mathematical thinking can help identify where the underlying difficulties begin.
Can a child have math gaps and still get good grades?
Yes. Some students compensate for foundational math gaps by memorizing formulas, following procedures, imitating examples, or recognizing familiar problem types. That can produce good grades without deep conceptual understanding. The gap may become more visible later when mathematics requires the student to transfer that knowledge, combine multiple concepts, or solve unfamiliar problems independently.
Can math gaps cause problems with algebra?
Absolutely. Algebra depends on earlier mathematical knowledge, including whole-number operations, multiplication and division, fractions, ratios, proportions, and equation concepts. A student may appear to have an “algebra problem” when an earlier foundational weakness is actually interfering with the algebra. Longitudinal research has found that earlier knowledge of fractions and division uniquely predicts later algebra achievement even after accounting for several other factors.
Are math gaps a sign of dyscalculia?
Not necessarily. Any student can develop a math gap because of missed instruction, incomplete understanding, or other factors. Dyscalculia is a persistent learning difficulty in mathematics that can affect different aspects of numerical and arithmetic learning, and its specific profile can vary considerably from one student to another. Students with dyscalculia may have more extensive or persistent foundational gaps, but having a math gap by itself does not mean a child has dyscalculia.
Can math learning gaps be fixed?
Math learning gaps can often be significantly remediated with appropriate, targeted instruction. The key is identifying the underlying concepts causing the difficulty rather than repeatedly practicing only the student's current grade-level struggle. For students with dyscalculia, ongoing specialized instruction and support may still be necessary even as specific gaps become much stronger.
What's the difference between math tutoring and math intervention?
Math tutoring often focuses on helping a student understand current coursework, while specialized math intervention investigates and remediates foundational concepts that may be preventing the student from accessing that coursework independently. A student struggling because of persistent foundational gaps or dyscalculia may need intervention that goes beneath the current chapter rather than additional practice with the chapter itself.
About Susan Ardila, M.Ed.
Susan Ardila, M.Ed., is an educational clinician, dyscalculia specialist, and founder of MindBridge Math Mastery, an online specialized math intervention practice serving students with persistent mathematics difficulties, dyscalculia, ADHD, and other learning differences.
With more than 12 years of experience in education, including classroom teaching, curriculum development, and individualized mathematics intervention, Susan specializes in identifying the foundational concepts underlying a student's current math difficulties and rebuilding mathematical understanding through explicit, multisensory, and conceptually focused instruction.
Her work emphasizes mathematical understanding, independence, and helping students move beyond memorized procedures toward mathematics that actually makes sense.
Research & References
The research and recommendations discussed in this article draw from the following peer-reviewed studies, systematic reviews, meta-analyses, and national evidence-based reports.
Castaldi, E., Piazza, M., & Iuculano, T. (2020). Learning disabilities: Developmental dyscalculia. Handbook of Clinical Neurology, 174, 61–75. https://doi.org/10.1016/B978-0-444-64148-9.00005-3
Cepeda, N. J., Pashler, H., Vul, E., Wixted, J. T., & Rohrer, D. (2006). Distributed practice in verbal recall tasks: A review and quantitative synthesis. Psychological Bulletin, 132(3), 354–380. https://doi.org/10.1037/0033-2909.132.3.354
Dunlosky, J., Rawson, K. A., Marsh, E. J., Nathan, M. J., & Willingham, D. T. (2013). Improving students' learning with effective learning techniques: Promising directions from cognitive and educational psychology. Psychological Science in the Public Interest, 14(1), 4–58. https://doi.org/10.1177/1529100612453266
Fuchs, L. S., Bucka, N., Clarke, B., Dougherty, B., Jordan, N. C., Karp, K. S., & Woodward, J. (2021). Assisting students struggling with mathematics: Intervention in the elementary grades. National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education.
Gersten, R., Chard, D. J., Jayanthi, M., Baker, S. K., Morphy, P., & Flojo, J. (2009). Mathematics instruction for students with learning disabilities: A meta-analysis of instructional components. Review of Educational Research, 79(3), 1202–1242. https://doi.org/10.3102/0034654309334431
Lafay, A., Osana, H. P., & Valat, M. (2019). Effects of interventions with manipulatives on immediate learning, maintenance, and transfer in children with mathematics learning disabilities: A systematic review. Education Research International, 2019, Article 2142948. https://doi.org/10.1155/2019/2142948
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. U.S. Department of Education.
National Research Council. (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, & B. Findell (Eds.). The National Academies Press. https://doi.org/10.17226/9822
Peltier, C., Morin, K. L., Bouck, E. C., Lingo, M. E., Pulos, J. M., Scheffler, F. A., Suk, A., Mathews, L. A., Sinclair, T. E., & Deardorff, M. E. (2020). A meta-analysis of single-case research using mathematics manipulatives with students at risk or identified with a disability. The Journal of Special Education, 54(1), 3–15. https://doi.org/10.1177/0022466919844516
Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., Susperreguy, M. I., & Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(7), 691–697. https://doi.org/10.1177/0956797612440101
Zhang, J., Zhao, N., & Kong, Q. P. (2019). The relationship between math anxiety and math performance: A meta-analytic investigation. Frontiers in Psychology, 10, Article 1613. https://doi.org/10.3389/fpsyg.2019.01613
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