1. Learning Disabilities in Mathematics 1
Learning Disabilities in Mathematics –
What High School Math Teachers Need to Know
William E. Engler
American International College

2. Learning Disabilities in Mathematics 2
A mathematics learning disability (MLD) is a specific type of learning disability that
affects about 5% of primary school students (Kaufmann, 2012). If diagnosed with a MLD, a
student is eligible for special education as per IDEA 2004. While no universally accepted
definition of a mathematics learning disability (MLD) exists, it is important to ground our
analysis with a definition that is both meaningful and helpful. At the risk of oversimplification, a
person with a MLD has a normal IQ but is unable to do math at the appropriate cognitive level,
based on his or her age and schooling. A learning disability can result from deficits in the ability
to represent or process information in one or all of the many mathematical domains, e.g.
geometry, or in one or a set of individual competencies within each domain. The most widely
used term for disabilities in arithmetic and mathematics is dyscalculia (Mazzocco, 2003).
Three subtypes of MLD have been proposed by Geary (1993). The first subtype is
characterized by procedural deficits. These individuals tend to use developmentally
unsophisticated procedures, make frequent errors, do not fully grasp the basic underlying
concepts, and exhibit sequencing difficulties. Persons with a procedural deficit have some form
of working memory deficiency. The overall performance of these individuals can be
summarized as developmentally delayed, as their performance in similar to that of younger
children. For example, individuals with procedural deficits may exhibit the following symptoms:
 Relatively frequent use of developmentally immature procedures.
 Frequent errors in the execution of procedures, e.g. does not distribute the negative
sign across a binomial; difficulty adding, subtracting, multiplying, dividing positive
and negative numbers; difficulty adding, subtracting, multiplying, dividing fractions;
difficulty applying rules for exponents; difficulty solving an equation with one or
more variables.

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 Poor understanding of the concepts underlying procedural use, e.g. does not relate a
negative number to the concept of a number line.
 Difficulties sequencing the multiple steps in complex procedures.
The second subtype is characterized by semantic memory deficits. These individuals have
difficulty retrieving math facts from long-term memory and there are often errors in the facts that
they are able to recall, suggesting that retrieval deficits resistant to instructional intervention
might be an indicator of mathematics learning disability. This usually coexists with a reading
disability. In contrast to a developmental delay, this second subtype is believed to represent a
qualitative difference in the cognitive processes underlying math aptitude. For example,
individuals with semantic memory deficits may exhibit the following symptoms:
 Difficulties retrieving mathematical facts, e.g., a high-school student may not have
memorized basic multiplication facts, how to convert a decimal number to a fraction
or percentage, how to interpret inequalities.
 For facts that are retrieved, there is a high error rate.
 For arithmetic, retrieval errors are often associates of numbers in the problem, e.g. a
student may say the square root of 16 is 8 (because 8 + 8 = 16) or 9 squared is 18
(because 9 + 9 = 18).
 Reaction times for correct retrieval are unsystematic.
The final subtype is characterized by poor spatial representation of numbers and other
mathematic information. These individuals have difficulty with properly aligning numeric
information, sign confusion, number omission or rotation, and general misinterpretation of
spatially relevant numerical information (e.g. place value). When writing, reading and recalling
numbers, these common mistakes occur: number additions, substitutions, transpositions,
omissions, and reversals.

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 Difficulties in spatially representing numerical and other forms of mathematical
information and relationships.
 Frequent misinterpretation or misunderstanding of spatially represented information.
For example, a student may have problems with textbook illustrations; with geometric
constructions; with diagrams of segments, planes, parallel lines, and vertical and
horizontal lines; confusion with graphs of functions; confusion with illustrations of
three-dimensional figures.
How do you know if a student has a MLD? Many professionals rely on standardized
achievement tests in combination with measures of IQ. Typical criteria for diagnosing a MLD
are scores lower than 20th
or 25th
percentile on math achievement tests combined with a low
average or higher IQ. Be aware that standardized achievement tests sample a broad range of
arithmetical and mathematical topics, whereas children with MLD often have severe deficits is
some areas and average or better competencies in others (Geary 1993).
Feifer (2007) provides a comprehensive list of purchasable assessments, categorized by
cognitive mechanisms. For example, assessments of working memory include Wechsler
Intelligence Scale for Children Fourth Edition Integrated (WISC IV Integrated), Stanford-Binet
Intelligence Scale Fifth Edition (SB5), Test of Memory and Learning; assessments of visual-
spatial functioning include WISC IV Integrated, SB5, Differential Ability Scales (DAS);
assessments of executive functions include Wisconsin Card Sort Test, NEPSY II, Behavior
Rating Inventory of Executive Functions; assessments of attention measures include Test of
Everyday Attention for Children (Tea-CH), NEPSY II, Cognitive Assessment System (CAS);
assessments of mathematical skills and number sense include Wechsler Individual Achievement
Test, Woodcock Johnson III Achievement Test, Test of Early Mathematics Ability; assessments
of math anxiety scales include Math Anxiety Rating Scale, State-Trait Anxiety Inventory;

5. Learning Disabilities in Mathematics 5
Achenbach Child Behavior Checklist. If you suspect a student has a math learning disability,
inform the appropriate special education professional at your school and discuss the possibility
of having the student evaluated.
As a classroom teacher, how can you accommodate students with math learning
disabilities? Baker et al (2009) synthesized findings from 42 interventions on instructional
approaches that improve the mathematics proficiency of students with learning disabilities. The
following instructional components resulted in significant mean effects: Explicit instruction,
visual representations, ranges of examples, student verbalizations, and providing ongoing
feedback.
In the most effective form of explicit instruction, the teacher demonstrated a step by step
plan (strategy) for solving the problem. The plan was problem specific (not a generic, heuristic
guide) and students were actively encouraged to use the same procedure/steps demonstrated by
the teacher.
Visual representations of problems that illustrate solution strategies should be used
frequently. For example, use number lines, color-coded graphs, three-dimensional figures, and
any visual context for word problems.
The teacher should thoughtfully plan the lesson by carefully selecting and sequencing
instructional examples. For example, for a lesson on inverse functions, the following
instructional examples could be presented in this order:
 Review the differences between a function and a relation.
 Review the vertical line test.
 Demonstrate how to find the inverse of a set of coordinate points by interchanging
the entries in each ordered pair.
 Define a one-to-one function.

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 Explain the horizontal line test.
 Explain how the domain and range of a function and its inverse are related.
 Have students graph a function and its inverse.
 Demonstrate how to find the inverse function f-1
(x) with an easy function.
 Demonstrate how to find the inverse function f-1
(x) with a hard function.
 Explain how to find the inverse of a domain-restricted function.
Teachers should encourage students to verbalize their thinking or their strategies, even if
the students were explicitly taught one specific way to solve a problem. Verbalization may help
to anchor skills and strategies both behaviorally and mathematically. While not a primary
instructional component, some heuristics trial and error could be explored with students with
MLDs.
Student math achievement is enhanced when teachers are provided with precise
information on student progress and specific areas of students’ strengths and weaknesses.
Schools should develop and implement progress monitoring systems in mathematics that include
graphs of student performance as well as specific instructional guidelines and curricular
materials for teachers or other relevant personnel to use with particular students.
It is still unclear whether or not the use of heuristics and peer-assisted instruction are
effective forms of instruction. When scaffolded to discover solutions on their own, students with
a MLD appear to arrive at a higher level of understanding. But the success of this approach may
be at odds with the notion that students with a MLD have difficulty with cognitively demanding
routines. The flexible use of heuristic strategies would seem to place a cognitive load on
students with a MLD that would make learning difficult. Also, within class peer-assisted
learning has not been as successful as it has with other populations.

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In summary, MLDs are recognized by IDEA 2004. MLD categories include procedural
deficits, semantic memory deficits, and weak spatial representation of numbers. Numerous
standardized tests have been developed that help teachers determine whether or not a student has
a MLD. Baker et al (2009) provides a list of effective accommodations that classroom teachers
can use immediately to help students with MLDs.

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References
Baker, S. K., Chard, D. J., Flojo, J., Gersten, R., Jayanthi, M., Morphy, P. (2009). Mathematics
Instruction for Students with Learning Disabilities: A Meta-Analysis of Instructional
Components. Review of Educational Research 79(3), 1202-1242.
Feifer, Steven G. (2007). The Neuropsychology of Math Disorders: Diagnosis and Intervention.
School Neuropsych Press.
Geary, David C. (2004). Mathematics and Learning Disabilities. Journal of Learning
Disabilities 37(1), 4-15.
Mazzocco, Michele M., Myers, Gwen F (2003). Complexities in Identifying and Defining
Mathematics Learning Disability in the Primary School-Age Years. Ann Dyslexia 53(1),
218-253.