By Betsy Hill and Roger Stark
Although reading difficulties have been a central focus of research for decades, math learning difficulties have received comparatively less attention. Yet math disabilities affect roughly 5 to 8 percent of the population, a prevalence similar to reading disabilities. Like reading problems, math difficulties can have multiple causes and present in different ways. Over the past decade or more the school math curriculum has shifted to emphasize communicating mathematical ideas, interpreting charts and graphs, and estimating, which means that weaknesses in a variety of cognitive areas can more easily undermine math performance. Many individuals with math disabilities also have co-occurring learning challenges, but regardless of comorbidity, evidence points to underlying cognitive processes as major contributors to math outcomes.
Research by Geary and others links specific cognitive mechanisms to distinct types of math difficulties. The following table summarizes these connections:
| Cognitive mechanisms | Math deficit |
|---|---|
| Language systems | Problems representing numerical information verbally, such as difficulty with number words and symbols |
| Working Memory | Trouble holding and manipulating information while performing operations, for example during counting or multi-step calculations |
| Visual-spatial processing | Difficulty forming spatial or magnitude representations, which affects understanding charts, place value and geometric relationships |
| Attentional and inhibitory processes (executive controls) | Challenges using and sequencing procedures accurately during problem solving |
From empirical findings linking these mechanisms to math performance, Geary identifies three common subtypes of math disability: Semantic Memory, Procedural, and Visual-Spatial. Below is an overview of each subtype, the typical difficulties involved, and the cognitive skills that underlie them.
Semantic Memory Subtype
Semantic memory supports our knowledge of words, symbols and concepts rather than memory for specific events. Students with semantic memory impairments often struggle to retrieve basic arithmetic facts quickly and reliably; they may be slow, inconsistent, or prone to errors when recalling simple sums or products. They may also fail to recognize operation symbols or understand differences between operations, which can lead to treating sequences of mixed-operation problems as if they were all the same type.
Because this subtype depends on phonological and semantic representations, it frequently co-occurs with language processing or reading disabilities.
Typical development of arithmetic fact knowledge progresses from concrete counting strategies to fast, automatic retrieval. For example, solving 4 + 6 usually moves from counting all, to counting on, to counting from the larger addend (minimization), and ultimately to direct retrieval from long-term memory. The final stage—automatic retrieval—usually requires explicit instruction and practice.
Traditional remediation emphasizes intensive drilling to store facts in long-term memory, but drill-based methods often fail for students with semantic memory weaknesses. Some educators rely on calculators or fact sheets to work around retrieval problems; while this can aid performance, it does not address the underlying cognitive deficit. More effective approaches aim to strengthen the memory and representational systems that support concept formation and symbolic retrieval rather than merely compensating for them.
Procedural Subtype
The Procedural subtype involves difficulties learning, sequencing, and executing multi-step procedures used in math. Affected students may use immature or inefficient strategies, make sequencing errors, or struggle to follow multi-step algorithms. These problems implicate sequential visual and auditory memory, working memory, logical reasoning and attention—skills necessary for recognizing patterns and reliably carrying out procedural steps.
Visual-Spatial Subtype
The Visual-Spatial subtype manifests as misalignment of columns in multi-digit arithmetic, digit reversals, difficulties with place value, or trouble interpreting spatial representations like exponents, charts and geometric figures. Foundational skills such as directionality, visualization, visual-spatial memory and rhythm/timing support these abilities and, when impaired, limit success in tasks that depend on spatial organization.
Core Deficits in Math Disabilities
While distinct subtypes are useful, research also points to common core deficits that cut across these categories. Identifying and targeting these underlying skills can guide more effective interventions. Key core deficits associated with poor math performance include:
● Strategy development
● Working memory
● Retrieval (automatic recall of facts)
● Conceptual knowledge
● Processing speed and fluency
● Language processing
● Counting knowledge
The following table details how specific cognitive skills support math tasks:
| Cognitive skill | How the skill relates to math |
|---|---|
| Visual Sustained Attention | Maintaining focus long enough to collect digits, symbols, or operators into the correct order for a calculation or expression |
| Visual Discrimination | Rapidly and accurately distinguishing signs and numerals (for example + versus × or 5 versus 2) and noticing important differences in charts and graphs |
| Visual Form Consistency | Recognizing an object or symbol regardless of size, orientation or distance, which supports geometric reasoning and symbol recognition |
| Visual Processing Speed | Faster processing of visual information enables grouping and integrating elements into meaningful mathematical structures |
| Visualization | Mentally representing and manipulating images—essential for understanding quantity changes, spatial relations and geometric transformations |
| Directionality | The ability to project left/right and spatial relationships onto objects supports number lines, place value and column alignment |
| Long-Term Memory | Storing and retrieving meanings, symbols and math facts is central to fluency and automaticity |
| Auditory/Visual Sequential Memory | Remembering the order of steps in a procedure ensures accurate execution of multi-step problems |
| Working Memory | Holding information while manipulating it—critical for combining facts, tracking interim results and following multi-step algorithms |
| Conceptual Thinking | Forming abstract categories like number, addition/subtraction and geometric concepts underpins higher-level math understanding |
The Cognitive Foundation for Counting and Number Sense
Although scholars debate the precise definition of number sense, most agree it is as fundamental to math as phonemic awareness is to reading. Number sense enables children to view numbers as discrete, ordered entities and to manipulate them automatically. When basic operations require conscious effort—such as counting on fingers for simple sums—cognitive resources are consumed and little remains for deeper problem solving. Developing number sense supports later calculation, estimation and problem solving.
Key components of number sense include:
● Counting
● Numerical relationships and magnitude comparison
● Number transformation (composing and decomposing numbers)
● Estimation
● Recognizing number patterns
● A mental number line
Counting itself is composed of several sub-skills, such as assigning number names, one-to-one correspondence (counting each item once), partitioning (keeping track of what has been counted), understanding cardinality (the last number named is the quantity), and abstraction (counting dissimilar or intangible items).
Research shows sensitivity to quantity emerges early in development, and some findings suggest timing or rhythm mechanisms support numerical discrimination. This observation helps explain links often observed between musical ability and math performance.
Historically, educators emphasized rote drill to build math fact automaticity. While drills can raise test performance, they often do little to develop robust number sense. A complementary strategy is to strengthen the underlying cognitive skills—visualization, visual form consistency, sequential memory and working memory—which supports both automaticity and concept formation and tends to yield more durable gains.
Some cognitive training programs target these foundational skills, improving working memory, processing speed and visual-spatial abilities that map directly onto the subtypes of math difficulty described above. Developing these core skills can be particularly valuable when math weaknesses are accompanied by other learning challenges.
Implications
Targeting basic cognitive skills offers a promising path to improve math performance, especially where traditional drill-based methods have produced limited results. Short-term, focused interventions that build working memory, processing speed, visualization and related skills can increase automaticity and free cognitive resources for deeper mathematical reasoning. Early identification and remediation of core deficits can therefore reduce the long-term academic impact of math learning difficulties and help learners make faster, more durable progress.
About the authors
Betsy Hill is President of BrainWare Learning Company, which applies neuroscience to improve learning capacity. An experienced educator, she has collaborated with experts in the science of learning and teaches strategic thinking in graduate programs. She holds an M.A. in Teaching and an MBA from Northwestern University and is co-author of the book Your Child Learns Differently, Now What?
Roger Stark is Co-founder and CEO of BrainWare Learning Company. He led development of integrated cognitive training tools intended to extend access to evidence-based cognitive skills training. He has focused on creating affordable, research-informed approaches to build cognitive literacy and is co-author of Your Child Learns Differently, Now What?