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What Is Dyscalculia? Math Learning Disability Overview

Dyscalculia is a learning disability that makes math challenging to process and understand. symptoms range from difficulty with counting and basic mental math to trouble with telling time and direction. learn more about this math learning disability, including potential causes and treatments here..

Dyscalculia Definition

Dyscalculia is a math learning disability that impairs an individual’s ability to learn number-related concepts, perform accurate math calculations, reason and problem solve, and perform other basic math skills. 1 Dyscalculia is sometimes called “number dyslexia” or “math dyslexia.”

Dyscalculia is present in about 11 percent of children with attention deficit hyperactivity disorder (ADHD or ADD). 2 Other learning disorders, including dyslexia and dysgraphia, are also common – up to 45 percent of children with ADHD have a learning disorder. 3

Dyscalculia Overview

Individuals with dyscalculia have difficulties with all areas of mathematics — problems not explained by a lack of proper education, intellectual disabilities, or other conditions. The learning disorder complicates and derails everyday aspects of life involving mathematical concepts – like telling time, counting money, and performing mental calculations.

“Students and adults with dyscalculia find math puzzling, frustrating, and difficult to learn,” says Glynis Hannell, a family psychologist and author of Dyscalculia: Action Plans for Successful Learning in Mathematics (#CommissionsEarned) . “Their brains need more teaching, more targeted learning experiences, and more practice to develop these networks.”

Dyscalculia frequently co-occurs with dyslexia , a learning disability in reading; about half of children with dyscalculia also have dyslexia. 4 While figures vary, the estimated prevalence of dyscalculia in school populations is 3 to 6 percent. 5

[ Take the Dyscalculia Symptom Test for Children ] [ Think You Have Dyscalculia? Take This Screener for Dyscalculia in Adults ]

Dyscalculia Symptoms

What are the signs of dyscalculia? Symptoms and indicators include 6 7 :

• Connecting a number to the quantity it represents (the number 2 to two apples)
• Counting, backwards and forwards
• Comparing two amounts
• Trouble with subitizing (recognize quantities without counting)
• Trouble recalling basic math facts (like multiplication tables)
• Difficulty linking numbers and symbols to amounts
• Trouble with mental math and problem-solving
• Difficulty making sense of money and estimating quantities
• Difficulty with telling time on an analog clock
• Poor visual and spatial orientation
• Difficulty immediately sorting out direction (right from left)
• Troubles with recognizing patterns and sequencing numbers

Finger-counting is typically linked to dyscalculia, but it is not an indicator of the condition outright. Persistent finger-counting, especially for easy, frequently repeated calculations, may indicate a problem.

Calculating errors alone are also not indicative of dyscalculia – variety, persistence, and frequency are key in determining if dyscalculia is present.

[ Watch: Early Warning Signs of Dyscalculia ]

Dyscalculia Causes

When considering dyscalculia, most people are actually thinking of developmental dyscalculia – difficulties in acquiring and performing basic math skills. Exact causes for this type of dyscalculia are unknown, though research points to issues in brain development and genetics (as the disability tends to run in families) as possible causes. 8

Acquired dyscalculia, sometimes called acalculia, is the loss of skill in mathematical skills and concepts due to disturbances like brain injury and other cognitive impairments. 9

Dyscalculia Diagnosis

Dyscalculia appears under the “specific learning disorder” (SLD) section in the Diagnostic and Statistical Manual of Mental Disorders 5th Edition (DSM-5). 10 For an SLD diagnosis, an individual must meet these four criteria:

• Individuals with dyscalculia exhibit at least one of six outlined symptoms related to difficulties with learning and using academic skills. Difficulties with mastering number sense and mathematical reasoning are included in the list.
• The affected academic skills are below what is expected for the individual’s age, which also cause trouble with school, work, or daily life.
• The learning difficulties began in school, even if problems only became acute in adulthood.
• Other conditions and factors are ruled out, including intellectual disabilities and neurological disorder, psychosocial adversity, and lack of instruction.

Individuals whose learning difficulties are mostly math-based may be diagnosed with “SLD with impairment in mathematics,” an SLD subtype equivalent to dyscalculia.

Diagnostic evaluations for dyscalculia are typically carried out by school psychologists and neuropsychologists, though child psychiatrists and school health services and staff may play a role in evaluation. Adults who suspect they have dyscalculia may be referred to a neuropsychologist by their primary care provider.

There is no single test for dyscalculia. Clinicians evaluate for the disorder by reviewing academic records and performance in standardized tests, asking about family history, and learning more about how the patient’s difficulties manifest in school, work, and everyday life. They may also administer diagnostic assessments that test strengths and weaknesses in foundational mathematical skills. Tools like the PAL-II Diagnostic Assessment (DA), the KeyMath-3 DA, and the WIATT-III are commonly used when evaluating for dyscalculia.

Dyscalculia Treatment and Accommodations

Like other learning disabilities, dyscalculia has no cure and cannot be treated with medication. By the time most individuals are diagnosed, they have a shaky math foundation. The goals of treatment, therefore, are to fill in as many gaps as possible and to develop coping mechanisms that can be used throughout life. This is typically done through special instruction, accommodations, and other interventions.

Under the Individuals with Disabilities Education Act ( IDEA ), students with dyscalculia are eligible for special services in the classroom. Dyscalculia accommodations in the classroom may include 11 :

• allowing more time on assignments and tests
• allowing the use of calculators
• adjusting the difficulty of the task
• separating complicated problems into smaller steps
• using posters to remind students to basic math concepts
• tutoring to target core, foundational skills
• computer-based interactive lessons
• hands-on projects

If left untreated, dyscalculia persists into adulthood, leaving many at a disadvantage when it comes to higher education and workplace success. 12 Adults with dyscalculia , however, may be entitled to reasonable accommodations in their workplace under the Americans with Disabilities Act ( ADA ). They can also commit to brushing up on math skills on their own or with the help of a trained educational psychologist. Even the most basic improvements in math skills can have long-lasting impacts on day-to day life.

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Dyscalculia At a Glance

Dyscalculia: next steps.

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View Article Sources

1 American Psychiatric Association. (2018, November). What is Specific Learning Disorder? https://www.psychiatry.org/patients-families/specific-learning-disorder/what-is-specific-learning-disorder

2 Soares, N., & Patel, D. R. (2015). Dyscalculia. International Journal of Child and Adolescent Health. https://psycnet.apa.org/record/2015-29454-003

3 DuPaul, G. J., Gormley, M. J., & Laracy, S. D. (2013). Comorbidity of LD and ADHD: implications of DSM-5 for assessment and treatment. Journal of learning disabilities, 46(1), 43–51. https://doi.org/10.1177/0022219412464351

4 Morsanyi, K., van Bers, B., McCormack, T., & McGourty, J. (2018). The prevalence of specific learning disorder in mathematics and comorbidity with other developmental disorders in primary school-age children. British journal of psychology (London, England : 1953), 109(4), 917–940. https://doi.org/10.1111/bjop.12322

5 Shalev, R.S., Auerbach, J., Manor, O. et al. Developmental dyscalculia: prevalence and prognosis. European Child & Adolescent Psychiatry 9, S58–S64 (2000). https://doi.org/10.1007/s007870070009

6 Haberstroh, S., & Schulte-Körne, G. (2019). The Diagnosis and Treatment of Dyscalculia. Deutsches Arzteblatt international, 116(7), 107–114. https://doi.org/10.3238/arztebl.2019.0107

7 Bird, Ronit. (2017). The Dyscalculia Toolkit. Sage Publications.

8 Szűcs, D., Goswami, U. (2013). Developmental dyscalculia: Fresh perspectives. Trends in Neuroscience and Education, 2(2),33-37. https://doi.org/10.1016/j.tine.2013.06.004

9 Ardila, A., & Rosselli, M. (2019). Cognitive Rehabilitation of Acquired Calculation Disturbances. Behavioural neurology, 2019, 3151092. https://doi.org/10.1155/2019/3151092

10 American Psychiatric Association (2014). Diagnostic and Statistical Manual of Mental Disorders. DSM-V. Washington, DC: American Psychiatric Publishing

11 N, Soares., Evans, T., & Patel, D. R. (2018). Specific learning disability in mathematics: a comprehensive review. Translational pediatrics, 7(1), 48–62. https://doi.org/10.21037/tp.2017.08.03

12 Kaufmann, L., & von Aster, M. (2012). The diagnosis and management of dyscalculia. Deutsches Arzteblatt international, 109(45), 767–778. https://doi.org/10.3238/arztebl.2012.0767

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Signs & Symptoms of Learning Disabilities

Dyscalculia

Affects a person’s ability to understand numbers and learn math facts..

Individuals with this type of learning disability demonstrate impaired math calculation skills and difficulty understanding numbers and math facts.

Dyscalculia is associated with weaknesses in fundamental number representation and processing, which results in difficulties with quantifying sets without counting, using nonverbal processes to complete simple numerical operations, and estimating relative magnitudes of sets.

Because these math skills are necessary for higher-level math problem solving, quantitative reasoning is likely impaired for these individuals.

Dyscalculia can impact:

• Estimating a quantity without counting
• Calculation skills
• Using processes to solve equations
• Mental math
• Remembering steps in a sequence
• Reading graphs or charts
• Remembering dates and deadlines
• Counting change
• Navigation skills

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

Dyscalculia

Table of Content

• What is dyscalculia

Etiology and Prevalence

Characteristics.

• Subtypes of Dyscalculia

Math Anxiety

• Intervention and Strategies

Assistive Technology

• LIving with Dyscalculia- video

Dyscalculia  (/ˌdɪskælˈkjuːliə/), is a disability resulting in difficulty learning or comprehending arithmetic, such as difficulty in understanding numbers, learning how to manipulate numbers, performing mathematical calculations and learning facts in mathematics. It is sometimes informally known as “math dyslexia”, though this can be misleading as dyslexia is a different condition from dyscalculia.

Dyscalculia does not reflect a general deficit in cognitive abilities or difficulties with time, measurement, and spatial reasoning. Estimates of the prevalence of dyscalculia range between 3 and 6% of the population. In 2015, it was established that 11% of children with dyscalculia also have ADHD.  Dyscalculia has also been associated with people who have Turner syndrome and people who have spina bifida.  (Wikipedia/dyscalculia, n.d)

A range of terms are used to refer to problems in learning mathematical concepts and skills, including Math Difficulties, Math Disability, Mathematical Learning Disability, Mathematics Disorder, Specific Disorder of Arithmetic Skills, Math Anxiety, and Developmental Dyscalculia ( DD). These terms are similar in that all implicate low numeracy skills. However, they are not synonymous  Here we differentiate general Mathematics Disorder [MD: e.g., DSM-IV:  from Developmental Dyscalculia in several important ways.

In Developmental Dyscalculia the learning problem:

1) is specific to the domain of arithmetic (reading and spelling skills are within the normal range);

2) manifests partly as problems in learning and remembering simple arithmetic facts (such as single-digit sums or products; e.g., 3+4 = 7), rather than more general problems in computation;

3) is typically defined by very low scores on standardized tests of arithmetic achievement, e.g., below the 8th or even 5th percentile, which is equivalent to standard scores below 78

4) reflects a specific impairment in brain function that gives rise to unexpected problems in basic numerical processing, such as automatic or implicit processing of quantities or numbers

(Rubinsten & Tannock, 2010)

• An important distinction between Developmental Dyscalculia and Mathematical deficits stemming from external factors.

Mathematical performance deficits, Developmental Dyscalculia, may arise because of a wide range of factors, from poor teaching to low socio-economic status, to behavioral attention problems. However, a subset of children with math difficulties, possibly with the most-severe impairments, appears to suffer from a developmental learning disorder that undermines the ability to process basic numerical magnitude information, and that impairment in turn undermines the acquisition of school-level arithmetic skills. This disorder, “primary developmental dyscalculia, ” should not be confused with “secondary developmental dyscalculia,” which refers to mathematical deficits stemming from external factors such as those described above. Instead, primary DD is associated with impaired development of brain mechanisms for processing numerical magnitude information and is thus driven by endogenous neurodevelopmental factors.  (Gaven & Ansari, 2013)

The term ‘dyscalculia’ was coined in the 1940s, but it was not completely recognized until 1974 by the work of Czechoslovakian researcher Ladislav Kosc. Kosc defined dyscalculia as “a structural disorder of mathematical abilities.” His research proved that the learning disability was caused by impairments to certain parts of the brain that control mathematical calculations and not because symptomatic individuals were ‘mentally handicapped’. Researchers now sometimes use the terms “math dyslexia” or “math learning disability” when they mention the condition. Cognitive disabilities, specific to mathematics were originally identified in case studies with patients who experienced specific arithmetic disabilities as a result of damage to specific regions of the brain. More commonly, dyscalculia occurs developmentally as a genetically linked learning disability which affects a person’s ability to understand, remember, or manipulate numbers or number facts (e.g., the multiplication tables). The term is often used to refer specifically to the inability to perform arithmetic operations, but is also defined by some educational professionals and cognitive psychologists such as  Stanislas Dehaene and  Brian Butterworth as a more fundamental inability to conceptualize numbers as abstract concepts of comparative quantities (a deficit in “number sense”), which these researchers consider to be a foundational skill upon which other mathematics abilities build. Symptoms of dyscalculia include the delay of simple counting, inability to memorize simple arithmetic facts such as adding, subtracting, etc. There are few known symptoms because little research has been done on the topic.  (Wikipedia/dyscalculia, n.d)

At its most basic level, dyscalculia is a learning disability affecting the normal development of arithmetic skills. A consensus has not yet been reached on appropriate diagnostic criteria for dyscalculia.

Other than using achievement tests as diagnostic criteria, researchers often rely on domain-specific tests (i.e. tests of working memory, executive function, inhibition, intelligence, etc.) and teacher evaluations to create a more comprehensive diagnosis. Alternatively, fMRI research has shown that the brains of the neurotypical children can be reliably distinguished from the brains of the children with dyscalculia based on the activation in the prefrontal cortex. (Wikipedia/dyscalculia, n.d)

Developmental Dyscalculia (DD) describes a specific and severe deficit in the ability to process numerical information that cannot be ascribed to sensory difficulties, low IQ or inadequate education, and that results in a failure to develop fluent numerical computation skills.  Untreated, DD typically persists beyond the school-age years into late adolescence and adulthood. Epidemiological studies indicate that DD is as common as reading disorders and affects 3.5% – 6.5% of the school-age population.  Moreover, DD runs in families and is heritable, which implicates genetic factors in its etiology, though to date, none have been reported,  (Rubinsten & Tannock, 2010)

The earliest appearance of dyscalculia is typically a deficit in  subitizing , the ability to know, from a brief glance and without counting, how many objects there are in a small group. Children as young as five can subitize six objects, especially looking at a die. However, children with dyscalculia can subitize fewer objects and even when correct take longer to identify the number than their age-matched peers. Dyscalculia often looks different at different ages. It tends to become more apparent as children get older; however, symptoms can appear as early as preschool. Common symptoms of dyscalculia are having difficulty with mental math, trouble analyzing time and reading an analog clock, struggle with motor sequencing that involves numbers, and often they will count on their fingers when adding numbers.

Dyscalculia is characterized by difficulties with common arithmetic tasks. These difficulties may include:

• Difficulty reading analog clocks.
• Difficulty stating which of two numbers is larger.
• Inability to comprehend financial planning or budgeting, sometimes even at a basic level; for example, estimating the cost of the items in a shopping basket or balancing a checkbook.
• Visualizing numbers as meaningless or nonsensical symbols, rather than perceiving them as characters indicating a numerical value (hence the misnomer, “math dyslexia”).
• Difficulty with multiplication, subtraction, addition, and division tables, mental arithmetic, etc.
• Inconsistent results in addition, subtraction, multiplication and division.
• When writing, reading and recalling numbers, mistakes may occur in the areas such as: number additions, substitutions, transpositions, omissions, and reversals.
• Poor memory (retention and retrieval) of math concepts; may be able to perform math operations one day, but draw a blank the next; may be able to do book work but then fails tests.
• Ability to grasp math on a conceptual level, but an inability to put those concepts into practice.
• Difficulty recalling the names of numbers, or thinking that certain different numbers “feel” the same (e.g. frequently interchanging the same two numbers for each other when reading or recalling them).
• Problems with differentiating between left and right.
• A “warped” sense of spatial awareness, or an understanding of shapes, distance, or volume that seems more like guesswork than actual comprehension.
• Difficulty with time, directions, recalling schedules, sequences of events, keeping track of time, frequently late or early.
• Difficulty reading maps.
• Difficulty working backwards in time (e.g. What time to leave if needing to be somewhere at ‘X’ time).
• Difficulty reading musical notation.
• Difficulty with choreographed dance steps.
• Having difficulty mentally estimating the measurement of an object or distance (e.g., whether something is 3 or 6 meters (10 or 20 feet) away).
• Inability to grasp and remember mathematical concepts, rules, formulae, and sequences.
• Inability to concentrate on mentally intensive tasks.
• Mistaken recollection of names, poor name/face retrieval, may substitute names beginning with the same letter..

(Wikipedia/dyscalculia, n.d)

[The National Center for Learning Disabilities],(2021, May 11). What is Dyscalculia, [Video file]. from https://youtu.be/HVf_OHK2hHQ (7:46 minutes)

Persistence in children

Although many researchers believe dyscalculia to be a persistent disorder, evidence on the persistence of dyscalculia remains mixed. [20]

Persistence in adults

There are very few studies of adults with dyscalculia who have had a history of it growing up, but such studies have shown that it can persist into adulthood. It can affect major parts of an adult’s life. Most adults with dyscalculia have a hard time processing math at a 4th grade level. For 1st-4th grade level, many adults will know what to do for the math problem, but they will often get them wrong because of “careless errors”, although they are not careless when it comes to the problem. The adults cannot process their errors on the math problems or may not even recognize that they have made these errors. Visual-spatial input, auditory input, and touch input will be affected due to these processing errors. People with dyscalculia may have a difficult time adding numbers in a column format because their mind can mix up the numbers, and it is possible that they may get the same (wrong) answer twice due to their mind processing the problem incorrectly. People with dyscalculia can have problems determining differences in different coins and their size or giving the correct amount of change and if numbers are grouped together, it is possible that they cannot determine which has less or more. If a person with dyscalculia is asked to choose the greater of two numbers, with the lesser number in a larger font than the greater number, they may take the question literally and pick the number with the bigger font. Adults with dyscalculia have a tough time with directions while driving and with controlling their finances, which causes difficulties on a day-to-day basis.

College students and other adult learners

College students, particularly may have a tougher time due to the fast pace and change in the difficulty of the work they are given. As a result of this, students may develop a lot of anxiety and frustration. After dealing with their anxiety for a long time, students can become averse to math and try to avoid it as much as possible, which may result in lower grades in math courses. However, students with dyscalculia often do exceptionally well in writing, reading, and speaking.  (Wikipedia/dyscalculia, n.d)

Research on subtypes of dyscalculia has begun without consensus; preliminary research has focused on comorbid learning disorders as subtyping candidates. The most common comorbidity in individuals with dyscalculia is dyslexia. Most studies done with comorbid samples versus dyscalculic-only samples have shown different mechanisms at work and additive effects of comorbidity, indicating that such subtyping may not be helpful in diagnosing dyscalculia. But there is variability in results at present.

Due to high comorbidity with other disabilities such as dyslexia and ADHD, some researchers have suggested the possibility of subtypes of mathematical disabilities with different underlying profiles and causes. Whether a particular subtype is specifically termed “dyscalculia” as opposed to a more general mathematical learning disability is somewhat under debate in the scientific literature.

• Semantic memory : This subtype often coexists with reading disabilities such as dyslexia and is characterized by poor representation and retrieval from long-term memory. These processes share a common neural pathway in the left  angular gyrus , which has been shown to be selective in arithmetic fact retrieval strategies and symbolic magnitude judgments. This region also shows low functional connectivity with language-related areas during phonological processing in adults with dyslexia. Thus, disruption to the left angular gyrus can cause both reading impairments and difficulties in calculation. This has been observed in individuals with Gerstmann syndrome, of which dyscalculia is one of a constellation of symptoms.
• Procedural concepts : Research by Geary has shown that in addition to increased problems with fact retrieval, children with math disabilities may rely on immature computational strategies. Specifically, children with mathematical disabilities showed poor command of counting strategies unrelated to their ability to retrieve numeric facts. This research notes that it is difficult to discern whether poor conceptual knowledge is indicative of a qualitative deficit in number processing or simply a delay in typical mathematical development.
• Working memory : Studies have found that children with dyscalculia showed impaired performance on working memory tasks compared to neurotypical children.  Working memory problems are confounded with domain-general learning difficulties, thus these deficits may not be specific to dyscalculia but rather may reflect a greater learning deficit. Dysfunction in prefrontal regions may also lead to deficits in working memory and other executive function, accounting for comorbidity with ADHD.  (Wikipedia/dyscalculia, n.d)

According to the data Rubinsten and Tannock, there is clear evidence that, for  Developmental Dyscalculia (DD), math words had an anxious influence mainly when it comes to addition and multiplication arithmetic problems. What may be some of the reasons for this phenomenon? Normally developing children enter school with informal knowledge about numbers and arithmetic; knowledge that is based on their daily experiences of counting and calculation. Once entering school, however, much educational training is focused on basic multiplication and addition arithmetic facts. Consider, however, a child with DD who is innately deficient in his/her ability to process numbers, to count and to calculate. This child, from a very young age, has to answer addition and multiplication questions for which there is almost always only one correct answer. This situation, combined with the culture of solving these problems quickly, can lead students with DD towards a negative attitude style and ultimately learned helplessness to arithmetic in general (i.e., the affectively related influence that negative affective words had on solving simple arithmetic problems). Also, this situation can lead to a specific and accentuated fear and avoidance when it comes to the retrieval of addition and multiplication problems from memory (i.e., the affectively related influence that math words had on solving mainly multiplication and addition problems).

Hembree  showed that cognitive-behavioral interventions for math anxiety had a positive influence on math achievement test scores. These findings are quite significant in terms of the relationship between math anxiety and math achievement, and specifically in relation to DD. For people with DD, childhood difficulties with numerical processes and poor math achievement intensify math anxiety, which further impedes math achievement. As educators come to appreciate the key role played by math anxiety, interventions that reduce it may become a key part of the math educational system. It might be that one of the most effective ways to reduce math anxiety is to improve math achievement from an early age through interventions focused on children with DD thus turning the cycle of failure-fear-failure to one of success-confidence-success. This is especially true if the assumption that DD is an innate condition is correct. Such programs would be an important way of helping students cope with the frustrations inherent in the learning of mathematics, and thereby improve math achievement.

Instructional strategies for students with Dyscalculia from DO-IT

Rochelle Kenyon lists the following strategies for teaching a student with math-related learning disabilities.

• Avoid memory overload. Assign manageable amounts of work as skills are learned.
• Build retention by providing review within a day or two of the initial learning of difficult skills.
• Provide supervised practice to prevent students from practicing misconceptions and “misrules.”
• Make new learning meaningful by relating the practice of subskills to the performance of the whole task.
• Reduce processing demands by preteaching component skills of algorithms and strategies.
• Help students visualize math problems by drawing.
• Use visual and auditory examples.
• Use real-life situations that make problems functional and applicable to everyday life.
• Do math problems on graph paper to keep the numbers in line.
• Use uncluttered worksheets to avoid too much visual information.
• Practice with age-appropriate games as motivational materials.
• Have students track their progress.
• Challenge critical thinking about real problems with problem solving.
• Use Manipulatives and technology such as tape recorders or calculators.

This list was adapted from the following source: Garnett, K., Frank, B., & Fleischner, J. X. (1983). A strategies generalization approach to basic fact learning (addition and subtraction lessons, manual #3; multiplication lessons, manual #5). Research Institute for the Study of Learning Disabilities. New York, NY: Teacher’s College, Columbia University.

Go to these resources for more interventions and strategies so support students with Dyscalulia

Manitoba.ca.(n.d.) Supporting Students with Mathematics Disability. Dyscalculia  Includes Characteristics of dyscalculia and lots instructional strategies.

Sharma, M. C. (2022), Some Remediation Principles of Dyscalculia and Acquired Dyscalculia,

Dyscalculia: Teaching Strategies and Modifications From [Teachings in Education]  (3:05 minutes)

{Teachings in Education (2020). Dyscalculia: Teaching Strategies & Modifications [Video File] From https://youtu.be/BWaam8s9wSs

For more math accommodations and teaching strategies to, consult some of the following online articles:

Kenyon, R, (2000) Accommodating Math Students with Learning Disabilities From https://www.ncsall.net/index.html@id=325.html

Manitoba.ca.(n.d.) Supporting Students with Mathematics Disability. From https://www.edu.gov.mb.ca/k12/docs/support/learn_disabilities/module5.pdf

Mathematics for Students with Learning Disabilities from Language-Minority Backgrounds: Recommendations for Teaching  by Diane Torres Raborn.

Assistive technology does not “cure” a specific learning disability. These tools compensate rather than remedy, allowing a person with an LD to demonstrate their intelligence and knowledge. Adaptive technology for the person with an LD is a made-to-fit implementation. Trial and error may be required to find a set of appropriate tools and techniques for a specific individual. Ideally, a person with an LD plays a key role in selecting their technology. The teacher should help to determine what works and what does not. Once basic tools and strategies are selected, they can be “test driven,” discarded, adapted, and/or refined.  (DO-IT, n.d.)

Living with Dsycalculia

[BBC The Social], (2019, Mar. 11), Living with Dyscalculia (It’s Not Just “Number Dyslexia” [Video File] From https://youtu.be/_djdPIZrFno  (3:04 minutes)

Extended Reading

Grigore, M. (2020). Towards a standard diagnostic tool for dyscalculia in school children.  CORE Proceedings ,  1 (1). https://doi.org/10.21428/bfdb1df5.d4be3454 https://core.pubpub.org/pub/ttoew31a/release/1

Rubinsten O (2015) Link between cognitive neuroscience and education: the case of clinical assessment of developmental dyscalculia. Front. Hum. Neurosci. 9:304. doi: 10.3389/fnhum.2015.00304  https://www.frontiersin.org/articles/10.3389/fnhum.2015.00304/full

Sharma, M, C, (2022). Some Remediation Principles for Dyscalculia adn Acquired Dyscalculia,  From https://www.bdadyslexia.org.uk/dyscalculia/professor-mahesh-c-sharma-full-article-bda-handbook-2022 

DO-IT, University of Washington, (2021) What are strategies for teaching a student with a math related learning disability? From https://www.washington.edu/doit/what-are-strategies-teaching-student-math-related-learning-disability (CC BY SA)

DO-IT, University of Washington, (2021), Why is accessible math important? From https://www.washington.edu/doit/why-accessible-math-important?3=   (CC BY SA)

Price, Gavin R., and Daniel Ansari. “ Dyscalculia: Characteristics, Causes, and Treatments. ” Numeracy 6, Iss. 1 (2013): Article 2. DOI: http://dx.doi.org/10.5038/1936-4660.6.1.2 (CC BY-NC 4.0)

Rubinsten, O., Tannock, R . Mathematics anxiety in children with developmental dyscalculia.   Behav Brain Funct   6,  46 (2010). https://doi.org/10.1186/1744-9081-6-46 (CC BY 2.0) https://behavioralandbrainfunctions.biomedcentral.com/articles/10.1186/1744-9081-6-46#citeas 

Wikipedia, (n.d.) Dyscalculia From https://en.wikipedia.org/wiki/Dyscalculia (CC SA)

updated 7/2/2023

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Cognitive Profiles of Mathematical Problem Solving Learning Disability for Different Definitions of Disability

Tammy d. tolar.

1 University of Houston, TX, USA

2 Vanderbilt University, Nashville, TN, USA

Jack M. Fletcher

Douglas fuchs, carol l. hamlett.

Three cohorts of third-grade students ( N = 813) were evaluated on achievement, cognitive abilities, and behavioral attention according to contrasting research traditions in defining math learning disability (LD) status: low achievement versus extremely low achievement and IQ-achievement discrepant versus strictly low-achieving LD. We use methods from these two traditions to form math problem solving LD groups. To evaluate group differences, we used MANOVA-based profile and canonical analyses to control for relations among the outcomes and regression to control for group definition variables. Results suggest that basic arithmetic is the key distinguishing characteristic that separates low-achieving problem solvers (including LD, regardless of definition) from typically achieving students. Word problem solving is the key distinguishing characteristic that separates IQ-achievement-discrepant from strictly low-achieving LD students, favoring the IQ-achievement-discrepant students.

There are at least two approaches to forming groups for research on children with math learning disabilities (LDs): comparing low achievement (LA) to extremely LA (e.g., Murphy, Mazzocco, Hanich, & Early, 2007 ) and comparing IQ-achievement discrepancy to LA only ( Fletcher et al., 1994 ). Typically groups formed according to these different definitions are compared on a variety of achievement and cognitive measures. The goal of these two approaches is to identify a group of students that is qualitatively distinct and warrants special treatment due to the disability of its members. If such a group exists, then identifying the students and the characteristics that distinguish them may lead to better methods for diagnosis and intervention.

In the math LD literature, LA-LD comparisons are more commonly evaluated than IQ-achievement comparisons. Numerical cognition, reading, working memory, processing speed, and visual-spatial processing have been implicated as potential indicators of qualitative differences between groups that differ in degree of low math achievement ( Geary, Hoard, Byrd-Craven, Nugent, & Numtee, 2007 ; Johnson, Humphrey, Mellard, Woods, & Swanson, 2010 ; Murphy et al., 2007 ). Comparisons in this literature have focused more on arithmetical skills than math problem solving, typically comparing LA (< 25th percentile, LA) to extremely LA (< 10th percentile, LD) as indicators of an LD. However, there is some variability in the cut points used to define LD status ( Johnson et al., 2010 ).

The IQ-achievement comparisons have been the subject of a relatively wide body of research, primarily for reading outcomes ( Fletcher et al., 1994 ). For reading LD, meta-analytic evidence based on a range of age groups, reading achievement definition measures, and external validation measures suggests that IQ-achievement-discrepant and LA-only LD students do not differ in achievement or behavior; there are small cognitive differences that can be explained by the relations between the cognitive and group definition variables ( Stuebing et al., 2002 ). These types of comparisons have not received much attention in the math LD literature. Mazzocco and Myers (2003) argue that the IQ-discrepant definition does not reliably discriminate between math disability (MD), borderline, and high-achieving students. However, an argument for evaluating the IQ-discrepant definition is that students who meet this criterion may be qualitatively different from students who are not discrepant but low achieving. If they are qualitatively different, then research on this group may provide some explanatory evidence related to MD. In addition, although states may not require IQ-achievement discrepancy criteria as the sole method for diagnosing an LD (per the Individuals With Disabilities Education Act, 2004, 34 CFR 300.309), many states do allow its use as one of several methods consistent with the Individuals With Disabilities Education Act 2004 (e.g., South Dakota Department of Education, 2013 ; Texas Education Agency, 2014 ).

In two of the few studies of MD that compared IQ-achievement-discrepant to LA-only LD students, González and Espinel ( 1999 , 2002 ) used a test of arithmetic as the definition variable on a group of 7- to 9-year-old students. The IQ-achievement-discrepant students were higher on performance and full IQ but not on verbal IQ. More specifically, the IQ-achievement-discrepant students were statistically higher than the LA group on IQ subtests of comprehension, vocabulary, picture completion, picture arrangement, block design, and object assembly but not arithmetic, similarities, information, and coding. The groups did not, however, differ statistically on working memory as measured by counting span, nor did they differ on word problem solving-performance (e.g., orally administered change, compare, combine, and equalize problems).

In these studies, students were assigned to one of the two comparison groups if their arithmetic achievement scores were below the 25th percentile. They were defined as IQ-achievement discrepant if their arithmetic achievement scores were more than 1 standard deviation lower than their IQ scores; they were defined as IQ-achievement consistent if their achievement scores were within 1 standard deviation of their IQ scores. Yet the achievement cutoff (25th percentile) was not consistent with more recent work that distinguishes students at or below the 10th percentile from students between the 10th and 25th percentiles ( Geary et al., 2007 ; Murphy et al., 2007 ). Also, the criteria used for establishing IQ-achievement discrepancy did not account for regression effects that are a consequence of the imperfect relation between IQ and arithmetic achievement (see Cahan, Fono, & Nirel, 2012 ; Fletcher et al., 1994 ; Stanovich & Siegel, 1994 , for extensive discussions on the consequences of not controlling for regression effects). Finally, it is not surprising that differences emerged on some of the IQ and math measures because they were used to define the groups.

At the same time, both traditions have come under scrutiny in terms of the potential for group differences on achievement or cognitive outcomes to be a consequence of statistical artifacts associated with creating cut points in univariate or multivariate normal distributional spaces—using one or more definitional variables and then comparing the groups on related variables ( Branum-Martin, Fletcher, & Stuebing, 2013 ). For example, if LD definitional groups are formed by dividing students based on levels of arithmetical performance, then the groups formed should differ on skills and abilities that are correlated with arithmetical skill. The magnitudes of expected differences correspond with the strengths of the relations between the skills/abilities and arithmetic performance. Similarly, if these groups are formed based on a combination of actual arithmetical performance and expected performance based on IQ, then the groups should differ on skills and abilities that are correlated with arithmetical skill and IQ. Branum-Martin et al. (2013) demonstrated these psychometric relations in a simulation study. Methods are available to control to some extent for these artifacts (e.g., controlling for the effects of the LD definition variables when evaluating group effects on external variables; see Stanovich & Siegel, 1994 ), but they have not been consistently applied to math LD in either research tradition.

According to Mazzocco and Myers (2003) , the cumulative nature of mathematics as well as qualitative differences in mathematics achievement across grade levels suggests the possibility that MD may manifest at different points and in different ways throughout development. Therefore, it is important to accumulate evidence regarding disability definitions across grades and math skills. One goal of this study was to add to the evidence by evaluating definitions of MD for math problem solving among third-grade students , which has not received as much research attention as calculation skills and students in lower grades.

We conducted two sets of group comparisons according to the two traditions (low achievement vs. extremely low achievement [LA-LD] and IQ-achievement-discrepant vs. strictly low-achieving LD [IQ-ACH]) among third-grade students evaluated on math-problem-solving achievement. To evaluate potential differences in achievement, we compared the groups on measures of math and reading achievement (math problem solving—nondefinition, word problem solving, arithmetic, computational fluency, word identification, sight word fluency, word reading, and passage comprehension). A math-problem-solving measure that was not used to define the LD groups (nondefinition measure) was included for external validation of the groups. LD groups should differ from non-LD groups but not from each other on a measure that is relatively closely aligned with the problem-solving definition measure. Computational fluency and arithmetic were included because these skills are manifestly required for math problem solving. In addition, computational fluency has been identified as a persistent problem for students with MDs ( Geary, 2004 ). Word problem solving was included because in some ways it aligns with math problem solving (e.g., problem solving applied to real-world scenarios); however, the numerical manipulation and computational requirements are not as great (at least in the context of this study).

We also compared the groups on cognitive abilities often differentially related to math difficulties ( Fuchs et al., 2008 ) including verbal IQ, nonverbal IQ, language, concept formation, long-term memory, working memory, short-term memory, processing speed, and attentive behavior. The reading achievement measures, cognitive abilities, and attentive behavior are all related to computational fluency and word problem solving among third-grade students ( Fuchs et al., 2006 ; Fuchs et al., 2008 ). Therefore, it was expected that students who perform poorly on measures of problem solving would also perform poorly on the reading and cognitive/behavioral measures.

In each set of comparisons, we evaluated effect sizes (i.e., univariate group comparisons without controlling for the group definition variables). We expected greater effect sizes for measures that are more highly correlated with the definition variables (e.g., greater group differences on math than reading measures between LD and non-LD groups, greater group differences on measures highly correlated with IQ between IQ-achievement-discrepant and LA-only students).

We also compared the groups using statistical controls to evaluate potentially misleading evidence as a consequence of statistical artifacts common to this type of research (i.e., groups are expected to differ on measures correlated with the group definition measures). We conducted regression analyses including the group and group definition variables as predictors of the external validation variables. This allowed us to evaluate whether there were group differences in achievement and cognitive abilities beyond what would be expected by the group performances on the correlated measures used to define the groups (i.e., problem solving and IQ). Finally, we conducted profile analyses based on MANOVAs to evaluate whether groups differed on achievement or cognitive variables beyond what would be expected given the correlations among the external validation measures (e.g., groups that differ on word reading would be expected to differ to some extent on reading fluency because these measures are correlated). Group differences beyond what would be expected given the correlations among the definition and external validation measures would suggest qualitative differences in the cognitive profiles.

We hypothesized that groups would not differ on most (and perhaps all) achievement and cognitive measures when controlling for the correlations among the measures. However, if there were group differences beyond what would be expected given the relations among the measures, we hypothesized these differences would be in computational fluency and working memory. These have been identified as particularly problematic for children with MDs ( Geary, 2004 ).

Participants

Participants were 813 third-grade students. The mean age of the students at the beginning of third grade was 8.5 years ( SD = 0.4 years). Of the students, 51% were female, 42% African American, 40% White, 12% Hispanic, 60% on free or reduced-price lunch, and 3% identified with an LD, Attention Deficit Hyperactivity Disorder (ADHD), or a speech, language, or hearing impairment.

These students were selected from three of four cohorts (957 students from consecutive years starting in 2004) on whom data was collected as part of a prospective 4-year study evaluating the effects of mathematics-problem-solving instruction and examining the developmental course and cognitive predictors of mathematics problem solving. One cohort was excluded because some measures evaluated in this study were not administered to that cohort.

Of the 957 students, 137 were excluded because they were missing one or more of the Grade 3 achievement, cognitive, or behavioral measures. The excluded students did not statistically differ from the participants on IQ, 95.2 vs. 97.3, F (1, 955) = 2.99, p > .05, or Woodcock-Johnson III (WJIII) Applied Problems , 100.8 vs. 103.0, F (1, 953) = 3.28, p > .05. Seven more students were excluded because performance on one of the measures was extremely low or high relative to their performance on other measures or to the rest of the sample (i.e., 1 student had an IQ score > 150, 3 students had Applied Problems scores more than 30 standard score points lower than their scores the other three times they were measured, 2 students had language standard scores less than 50 although their IQ and problem-solving scores were average, and 1 student had a visual matching score less than 20 standard score points). Based on achievement and cognitive test performance (see Table 1 ), the 813 students are a representative sample of third-grade students (e.g., Applied Problems and IQ scores and standard deviations are consistent with average performance based on test norming samples; see Table 1 ).

Means, Standard Deviations, and Correlations With Learning Disability Definition Measures ( N = 813).

Note . All means except math problem solving–nondefinition, word problem solving, computational fluency, verbal and nonverbal IQ, and attentive behavior are based on age-normed standard scores ( M = 100, SD = 15). The math-problem-solving (nondefinition) mean is based on grade-normed standard scores where the mean performance among Grade 3 students tested in the spring is 185. The word-problem-solving mean is based on raw scores with possible scores from 0 to 33. The computational fluency mean is based on raw problems per minute with a maximum possible score of 19.4. The verbal and nonverbal IQ means are based on age-normed T-scores ( M = 50, SD = 10). The attentive behavior mean is based on raw scores with possible scores ranging from −27 to 27 (0 represents a teacher’s reported rating of “average” attentive behavior). MPS-def = math problem solving–definition. All correlations are significant, p < .05.

Achievement Measures

Math problem solving–definition.

WJIII Applied Problems ( Woodcock, McGrew, & Mather, 2001a ) is used to define LD status in this study. Applied Problems is a 60-item test that measures skill in analyzing and solving practical math problems such as counting and telling time or temperature. Items are presented orally, and testing is discontinued after six consecutive errors. Internal consistency and test-retest reliability estimates are r > .90 ( Alfonso & Flanagan, 2002 ) and r = .85 ( McGrew & Woodcock, 2001 ) for children in the age range of study participants. Age-normed standard scores ( M = 100, SD = 15) are used for this study.

Math problem solving–nondefinition

Iowa Test of Basic Skills (ITBS): Problem Solving and Data Interpretation ( Hoover, Dunbar, & Frisbie, 2001 ) measures skill in solving arithmetical word problems and using data presented in tables and graphs to solve word problems. Internal consistency estimates for Grades 1 to 5 are .83 to .87. The mean grade-based standard score for third-grade students tested in the spring is 185 ( University of Iowa, 2014 ). The average of standard scores from Forms A and K are used for this study.

Word problem solving

Word-problem-solving scores are based on the sum of scores from Story Problems ( Jordan & Hanich, 2000 ) and Algorithmic Word Problems ( Fuchs, Hamlett, & Powell, 2003 ). Story Problems includes 14 brief story problems involving sums or minuends of 9 or less, with change, combine, compare, and equalize relationships. The tester reads each item aloud; students have 30 s to respond and can ask for rereading(s) as needed. The score is the number of correct answers. Estimated internal consistency is .86 ( Fuchs et al., 2008 ). Algorithmic Word Problems includes 9 problems, each of which requires one to four steps. Each problem conforms to a single problem type. Research assistants read aloud each item while students follow along on their own copies of the problems. Problems are scored according to a rubric that assigns points for correct intermediate calculations and final solutions. The maximum score is 19. Estimated internal consistency is .85 ( Fuchs et al., 2008 ).

The Wide Range Achievement Test-Third Edition Arithmetic Subtest ( Wilkinson, 1993 ) measures skills in counting, number identification, number comparison, and written computations. Internal consistency is high ( r > .90). Age-normed standard scores ( M = 100, SD = 15) are used for this study.

Computational fluency

Computational fluency scores are based on weighted averages of three assessments: the Addition Fact Fluency and Subtraction Fact Fluency subtests from the Grade 3 Math Battery ( Fuchs et al., 2003 ) and the Second Grade Test of Computational Fluency ( Fuchs, Hamlett, & Fuchs, 1990 ). The Addition and Subtraction Fact Subtests each comprise 25 problems with answers from 0 to 12, and students have 1 min to write answers. Estimated coefficient alpha among third-grade students is .92 ( Fuchs et al., 2008 ). The Second Grade Test of Computational Fluency ( Fuchs et al., 1990 ) is a single page of computation items representing the second-grade curriculum: seven single-digit addition items, seven single-digit subtraction items, three double-digit addition items without regrouping, two double-digit addition problems with regrouping, three double-digit subtraction items without regrouping, and three double-digit subtraction items with regrouping. Items are displayed in five rows of 5 items, and students have 3 min to write answers next to or below each item. Estimated coefficient alpha among third-grade students is .94 ( Fuchs et al., 2008 ). Across the three computational fluency measures, the focus is primarily on retrieval of math facts; 11 out of 75 problems across the three measures require double-digit computations, but only 5 of those require regrouping. We computed a problem-correct-per-minute score by averaging a third of the Computational Fluency score with the Addition and Subtraction Fact fluency scores. The maximum score possible is 19.4 problems correct per minute.

Word identification

The Woodcock Reading Mastery Test-Revised, Word Identification ( Woodcock, 1998 ) measures real-world reading ability with 100 words arranged in order of difficulty. Students read words aloud. Testing is discontinued after six consecutive errors at the end of a page. As per Woodcock, split-half reliability is .98. Age-normed standard scores ( M = 100, SD = 15) are used for this study.

Sight word fluency

Test of Word Reading Efficiency—Sight Word Efficiency ( Torgesen, Wagner, & Rashotte, 1999 ) measures the number of real words students can read in 45 s. As per the test developer, reliability is .95 for 8-year-olds. Age-normed standard scores ( M = 100, SD = 15) are used for this study.

Word reading

The Wide Range Achievement Test-Third Edition, Reading Subtest ( Wilkinson, 1993 ) measures skills in letter identification and word recognition. Internal consistency is high (> .90). Age-normed standard scores ( M = 100, SD = 15) are used for this study.

Passage comprehension

Woodcock Reading Mastery Test-Revised, Passage Comprehension ( Woodcock, 1998 ) measures students’ ability to identify a picture corresponding to symbols and words and to identify missing words in passages. Split-half reliability exceeds .90. Age-normed standard scores ( M = 100, SD = 15) are used for this study.

Cognitive/Behavioral Measures

The Wechsler Abbreviated Scale of Intelligence (WASI; Wechsler, 1999 ) Full Scale IQ score is used to define LD status in this study. The Full Scale IQ is based on two subtests: Vocabulary and Matrix Reasoning (see below). Split-half reliability is .95 for Full Scale IQ ( Zhu, 1999 ). Correlations between IQ based on Vocabulary and Matrix Reasoning alone and IQ based on all four subtests is r > .93 ( Wechsler, 2003 ). Age-normed standard scores ( M = 100, SD = 15) are used for this study.

WASI Vocabulary ( Wechsler, 1999 ) measures expressive vocabulary, verbal knowledge, and foundation of information with 42 items. The first 4 items present pictures; the student identifies the object in the picture. For the remaining items, the tester says a word that the student defines. Responses are awarded a score of 0, 1, or 2, depending on the quality of response. Testing is discontinued after five consecutive scores of 0. As per the test developer, based on a standardization sample representative of the U.S. English-speaking population, split-half reliability coefficients are estimated to be .86 and .88 for 8- and 9-year-olds, and test-retest reliability coefficients are estimated to be .84 for 6- to 11-year-olds. Age-normed T-scores are used in all analyses with expected M ( SD ) = 50 (10) for the representative sample described above.

Nonverbal IQ

WASI Matrix Reasoning ( Wechsler, 1999 ) measures nonverbal reasoning with four types of tasks: pattern completion, classification, analogy, and serial reasoning. Examinees look at a matrix from which a section is missing and complete the matrix by selecting among five options. Testing is discontinued after four errors on five consecutive items or after four consecutive errors. As per the test developer, based on a standardization sample representative of the U.S. English-speaking population, split-half reliability coefficients are estimated to be .94 and .93 for 8- and 9-year-olds, and test-retest reliability coefficients are estimated to be .77 for 6- to 11-year-olds. Age-normed T-scores are used in all analyses with expected M ( SD ) = 50 (10) for the representative sample described above.

Language scores are based on the average of Test of Language Development Grammatic Closure ( Newcomer & Hammill, 1988 ) and Woodcock Diagnostic Reading Battery Listening Comprehension ( Woodcock, 1997 ). Grammatic Closure measures the ability to recognize, understand, and use English morphological forms. The examiner reads sentences, one at a time; each sentence has a missing word. As reported by the test developer, reliability is .88 for 8-year-olds. Listening Comprehension measures the ability to understand sentences or passages. Students supply the word missing from the end of each orally presented sentence or passage. Reliability is .80 at ages 5 to 18. Age-normed standard scores ( M = 100, SD = 15) are used in this study.

Concept formation

WJIII Concept Formation ( Woodcock, McGrew, & Mather, 2001b ) measures the ability of students to identify rules for concepts when shown illustrations of instances and noninstances of the concept. As per the test developer, reliability is .93 for 8-year-olds. Age-normed standard scores ( M = 100, SD = 15) are used in this study.

Long-term memory

WJIII Retrieval Fluency ( Woodcock & Johnson, 1989 ) measures long-term memory by asking examinees to recall related items, within categories, for 1 min per category. As reported by the test developer, reliability is .78 for 8-year-olds. Age-normed standard scores ( M = 100, SD = 15) are used in this study.

Working memory

Working Memory Test Battery for Children, Listening Recall ( Pickering & Gathercole, 2001 ) measures students’ ability to recall the final words of 1 to 6 sentences after hearing all sentences and determining whether the sentences are true. As per Pickering and Gathercole (2001) , test-retest reliability is .93. Age-normed standard scores ( M = 100, SD = 15) are used in this study.

Short-term memory

WJIII Numbers Reversed ( Woodcock & Johnson, 1989 ) measures the ability to repeat in reverse order a string of orally presented random numbers. Numbers reversed was evaluated separately from listening recall because evidence suggests that numbers reversed may not represent the same construct as listening recall among older children. Beginning at 8 or 9 years, reverse digit span does not load as highly on a working memory latent factor as listening recall (.50 vs. .69; Gathercole, Pickering, Ambridge, & Wearing, 2004 ). By 10 to 12 years, when reverse digit span is included in a working memory factor, working memory correlates > .90 with a short-term memory factor that includes forward span measures. The two factors based on such a measurement model may represent a single construct. In addition, the different encoding requirements (i.e., numbers vs. words) may cause numbers reversed and listening recall to uniquely discriminate groups defined by levels of math achievement ( Dark & Benbow, 1991 ; Henry, 2001 ). Numbers reversed is labeled short-term memory in this study because this is how some researchers label it ( Rosen & Engle, 1997 ); however, it may represent more than one construct. As per the test developer, reliability for numbers reversed is .86 for 8-year-olds. Age-normed standard scores ( M = 100, SD = 15) are used in this study.

Processing speed

WJIII Visual Matching ( Woodcock et al., 2001b ) asks examinees to locate and circle two identical numbers that appear in a row of six numbers; examinees have 3 min to complete 60 rows. As per the test developer, reliability is .91 for 8-year-olds. Age-normed standard scores ( M = 100, SD = 15) are used in this study.

Attentive behavior

The SWAN Rating Scale ( Swanson et al., 2004 ) is an 18-item teacher rating scale, 9 of which items assess the inattentive behavior criteria for ADHD from the Diagnostic and Statistical Manual of Mental Disorders (4th ed.; American Psychiatric Association, 1994 ). The items are evaluated on a 7-point scale centered at 0 (range = −3 to 3). For this study the items are reverse scored to represent attentive behavior. The summary score is the sum across the 9 items, ranging from −27 to 27 points. Scores less than −22 would be considered clinically significant for ADHD (derived from 2.48 cutoff for mean item scoring; see Swanson et al., 2004 ). Internal consistency is estimated to be .98 ( Tolar et al., 2012 ).

Test administration

All math achievement tests except WJIII Applied Problems (the LD definition measure) and ITBS Problem Solving were administered in September in whole-class sessions. ITBS Problem Solving was administered in whole-class sessions in early March. WJIII Applied Problems , all reading achievement tests except WJIII Passage Comprehension , and all cognitive tests were administered during September and October in individual test sessions by trained research assistants according to standard instructions. WJIII Passage Comprehension was administered in individual sessions in late March and early April. The individual test sessions were audio-taped, and 17.9% of tapes, distributed equally across testers, were selected randomly for accuracy checks by an independent scorer. Agreement estimates exceeded 98.8%. In October, teachers completed the SWAN Rating Scale on each student.

LD group classifications

Students were classified for two types of comparisons: (a) LA versus LD (LA-LD) comparisons and (b) IQ-ACH comparisons.

LA-LD comparison groups

We used sample-based percentiles to classify students into learning disabled, low achieving, and no LD. The sample of Grade 3 students in this study was substantially larger than the one used for norming WJIII Applied Problems (813 vs. 490; McGrew, Schrank, & Woodcock, 2007 ) and differed in demographic makeup (e.g., 42% vs. 15% African American). The sample-based cutoffs for the 10th and 25th percentile were 87 and 94, respectively, whereas the test norm cutoffs are 80 and 90, suggesting the study sample has greater problem-solving skills than the test-norming sample. For this study, if students scored less than or equal to a standard score of 87 on math problem solving-definition they were classified as learning disabled. If they scored greater than 87 but less than or equal to 94 they were classified as low achievers. If they scored greater than 94 they were classified as having no LD. Based on these definitions, 74 students were classified as learning disabled, 134 as low achieving, and 605 as not learning disabled.

IQ-ACH comparison groups

We used the methods described in Fletcher et al. (1994) to identify students with discrepancies between their math-problem-solving performance and their predicted performance based on IQ. The sample-based correlation between IQ and WJIII Applied Problems ( r = .60) was used to calculate predicted math-problem-solving scores and standard errors of prediction. The correlation between IQ and math reasoning (WJIII Applied Problems ) among the test-norming samples ranged from .61 to .73 depending on the IQ measure and age group (6-8 and 9-13 years, correlations are lower among the younger children; McGrew et al., 2007 ). This suggests that math-problem-solving achievement may be slightly less related to IQ among the study students. For this study, students whose actual math-problem-solving scores were less than or equal to 1.5 standard errors of prediction below predicted scores were classified as learning disabled based on an IQ-achievement discrepancy. Students who were not classified as IQ-achievement discrepant but scored less than or equal to 87 (10th percentile) on math problem solving were classified as learning disabled based on achievement alone. According to these definitions, 42 students were classified as learning disabled due to an IQ-achievement discrepancy, 42 as learning disabled based on achievement alone, and 729 as not learning disabled. Among the IQ-achievement-discrepant students, 32 (76%) also met criteria for math-problem-solving LD based on achievement alone.

Analytic procedures

For each set of comparison groups, three types of analyses were conducted: profile, canonical, and regression. All analyses were conducted separately on two sets of outcomes: achievement and cognitive/behavioral. The same achievement measures were used in both sets of group comparisons; however, for the cognitive/behavioral measures, verbal and nonverbal IQ were included in the LA-LD but not the IQ-ACH comparisons because they were nondefinition variables for the former but not the latter.

Profile analyses

Profile analysis is based on MANOVA and typically involves tests for shape, level, and flatness. Shape effects (group by outcome interactions) indicate different profiles (i.e., qualitative differences) and are the focus of this study. In the event of no shape effect, level or group main effects indicate group differences across outcomes. Flatness is indicated by main effects of outcome (i.e., across groups, scores are higher on one outcome than another). Any potential flatness effects were removed by using z-scores (calculated on the full sample of 813 students) in the MANOVA analyses. For the profile analyses, we included the no-LD as well as the LD groups from each set of comparison groups.

Canonical analyses

Canonical analysis is derived from MANOVA, which computes a linear function of the outcome variables that maximizes the difference between the groups being evaluated. It permits interpretation of the discriminant functions based on the weightings of the variables making up the discriminant function ( Huberty & Olejnik, 2006 ). We evaluated both the standardized discriminant function coefficients, which represent the relative weights of the variables in the discriminant functions, and the canonical correlations, which represent the absolute relations between the discriminant functions and the outcome variables. For this set of analyses we included only the LD groups (i.e., LA-LD: low achieving vs. learning disabled; IQ-ACH: learning disabled based on IQ discrepancy vs. learning disabled based on LA alone).

Regression analyses

Regression analyses were conducted for each outcome. Each regression model included the LD definition variables: math problem solving for the LA-LD comparison groups and math problem solving and IQ for the IQ-ACH comparison groups (see Stanovich & Siegel, 1994 , for detailed explanation and application of this method to reading disabilities). Groups formed from the division of continuous, normally distributed measures are more likely to differ on measures that are more highly correlated with the definition measures. This is a statistical artifact. By controlling for the definition variables, we are evaluating whether individuals at similar levels of problem solving and IQ performance differ by LD status on the outcome variables.

All regression models were first evaluated with just the definition variables to determine if there were nonlinearities. Nonlinearities were identified between math problem solving-nondefinition and math problem solving-definition, word problem solving and IQ, and word identification and IQ. Therefore, for the LA-LD comparison group regression analysis involving math problem solving-nondefinition, a squared term for math problem solving-definition was included in the model. For IQ-ACH comparison group analyses involving word problem solving and word identification, a squared term for IQ was included in the models.

Two contrast variables were evaluated in the regression models: The no-LD groups versus the LA-LD or IQ-ACH groups and LA versus LD or IQ versus ACH groups. This allowed us to evaluate whether there is an additional effect of LD status on the outcome variable beyond general low mathematical-problem-solving status.

Evaluation of Cohorts

The three cohorts of students did not statistically differ in group assignments, LA-LD, χ 2 (4) = 1.71, p > .05; IQ-ACH, χ 2 (4) = 1.59, p > .05. MANOVA was used to compare the cohorts on all measures used in the study. There was a significant measure by group interaction, F (32, 12960) = 2.49, p < .05. However, the significant effect is likely due to the large group sizes ( n > 258 per group) and the large number of variables. The differences did not appear to be practically significant. One effect size was .39 (sight word fluency, standard scores of 105.1 and 100.5 for Cohorts 3 and 4, respectively). The rest of the effect sizes were less than .30. There was no systematic pattern in the cohort differences. For these reasons, all subsequent analyses were collapsed across cohorts.

Effect Sizes

Raw effect sizes were calculated to evaluate the magnitude of group differences without controlling for the relations between the group definition variables and the achievement, cognitive, and behavioral variables.

Definition variables

The largest differences between groups were, as expected, in the definition variables (see Table 2 ). For the LA-LD comparison groups, the effect sizes (i.e., number of standard deviations difference) for math problem solving-definition were 1.36 (no-LD vs. LA), 2.12 (no-LD vs. LD), and 0.76 (LA vs. LD). For the IQ-ACH comparison groups, there was a small difference in IQ between no-LD students and IQ-achievement-discrepant LD students (effect size = 0.25; see Table 2 ) and a small difference in math-problem-solving achievement (definition) between IQ-achievement-discrepant and LA-only (ACH) LD students (effect size = −0.18). Otherwise, for the IQ-ACH comparison groups, the effect sizes were large when comparing no-LD students to IQ and ACH students on math problem solving-definition (1.87 and 1.69; see Table 2 ), no-LD students to ACH students on IQ (1.22), and IQ to ACH students on IQ (0.97).

Effect Sizes Based on Different Definitions of Learning Disability (LA-LD, IQ-ACH).

Note . Effect sizes are based on raw or standard scores (see Measures section for scoring) and were computed as the difference in-group mean scores divided by the sample ( n = 813) standard deviations. Positive effect sizes favor the first group (e.g., for LA vs. LD, LA is the first group, and LD is the second group). For LA versus LD and IQ versus ACH comparisons, large effect sizes (> .70) are bolded. LA-LD = low achievement versus extremely low achievement; IQ-ACH = IQ-achievement discrepant versus strictly low-achieving LD; NO LD = students do not meet criteria for low achieving or learning disabled; LA = low achieving in math problem solving (≤ 25th percentile, > 10th percentile), LD = learning disabled in math problem solving based on low achievement (≤ 10th percentile), IQ = LD based on discrepancy between IQ and mathproblem solving achievement; ACH = LD based on low achievement in math problem solving only.

Nondefinition variables

Overall, there were large differences (effect sizes > 0.60; see Table 2 ) in achievement, cognitive abilities, and attentive behavior when comparing no-LD students to low-achieving or LD students (regardless of the way the LD groups were defined) with two general exceptions. First, the differences in memory (long term, working, and short term) between no-LD and low-achieving students (LA-LD comparison groups) were more moderate than differences on other achievement and cognitive measures (effect sizes = 0.54, 0.55, and 0.60). Second, the differences in cognitive abilities were small to moderate (effect sizes 0.19-0.49) between no-LD and IQ-discrepant students except for concept formation (0.65), processing speed (0.80), and attentive behavior (0.63).

In general, the differences in achievement, cognitive abilities, and attentive behavior were low to moderate (see Table 2 ) between low-achieving and LD students (LA-LD definitional groups) and between IQ-discrepant LD and LA-only LD students (IQ-ACH definitional groups). The largest effect size for the LA versus LD comparison was in arithmetic (0.69) and for IQ versus ACH comparison was in working memory (0.58).

Profile Analyses

There were LA-LD group differences in profile shape for the achievement but not the cognitive/behavioral outcomes as indicated by a statistically significant achievement by group interaction, F (14, 5670) = 4.16, p < .05, but nonsignificant cognitive/behavioral outcome by group interaction, F (16, 6480) = 1.18, p > .05. The achievement shape differences appear to be primarily between the LA and LD groups. Based on post hoc analyses, LA students scored significantly higher than LD students on all math measures, word reading, and passage comprehension ( p < .05) but not word identification or sight word reading fluency (see Table 3 ). The no-LD students scored significantly higher than both the LA and LD students on all achievement measures.

MANOVA and Regression Results (LA-LD Definitional Group Comparisons).

Note . LA-LD = low achievement versus extremely low achievement; mps-nd = math problem solving-nondefinition; wps = word problem solving; arith = arithmetic; comp = computational fluency; wi = word identification; sw = sight word fluency; rd = word reading; pc = passage comprehension; NO LD = students do not meet criteria for low achieving or learning disabled; LD = learning disability; LA = low achievement; sdfc =standardized discriminant function coefficient; r = canonical correlation; viq = verbal IQ; nviq = nonverbal IQ; lang = language; cf = concept formation; ltm = long-term memory; wm = working memory; stm = short-term memory; ps = processing speed; att = attentive.

There were level differences in cognitive abilities between LA-LD groups as indicated by a statistically significant main effect of group, F (2, 810) = 149.46, p < .05. These differences appear to be primarily between the no-LD students and the two lower achieving groups (LA and LD). Based on post hoc comparisons, the no-LD students scored statistically higher than both low-achieving groups on all cognitive/behavioral measures ( p < .05; see Table 3 ) whereas the LA and LD students did not statistically differ on any measure with the exception of concept formation and working memory.

The LA-LD shape effects in the achievement measures and level effects in the cognitive/behavioral measures can be seen in Figure 1 . There is a slightly larger gap (approximately .5 or greater z score) among the math than the reading measures between the LA and LD students whereas there is a relatively small gap (approximately .25 or less z score) across all cognitive/behavioral measures between the two groups of students. The difference between the no-LD and lower achieving groups was consistently large (approximately .75 or greater z score) across all achievement and cognitive/behavioral measures.

Achievement (a) and cognitive (b) profiles of LA-LD definitional groups.

Note. LA-LD = low achievement versus extremely low achievement; mps-nd = math problem solving-nondefinition; wps = word problem solving; arith = arithmetic; comp = computational fluency; wi = word identification; sw = sight word fluency; read = word reading; pc = passage comprehension; viq = verbal IQ; nviq = nonverbal IQ; lang = language; cf = concept formation; ltm = long-term memory; wm = working memory; stm = short-term memory; ps = processing speed; att = attentive behavior.

Similar to the LA-LD group comparisons, there were differences between the IQ-ACH comparison groups in profile shapes for the achievement but not the cognitive/behavioral outcomes as indicated by a statistically significant achievement by group interaction, F (14, 5670) = 3.58, p < .05, but nonsignificant cognitive/behavioral outcome by group interaction, F (12, 4860) = 1.33, p > .05. Based on post hoc analyses, IQ-discrepant LD students scored significantly higher than low-achieving-only LD students on word problem solving ( p < .05) but none of the other achievement measures (see Table 3 ). The no-LD students scored significantly higher than both the IQ-discrepant and low-achieving-only LD students on all achievement measures.

There were level differences in cognitive abilities across the IQ-ACH groups as indicated by a statistically significant main effect of group, F (2, 810) = 46.33, p < .05. Similar to the LA-LD groups, these differences appear to be primarily between the no-LD students and the two LD groups. Based on post hoc comparisons, the no-LD students scored statistically higher than both LD groups on all cognitive/behavioral measures ( p < .05; see Table 3 ) whereas the IQ-discrepant and low-achieving-only LD students did not differ statistically on any measure with the exception of working memory.

The IQ-ACH shape effects in the achievement measures and level effects in the cognitive/behavioral measures can be seen in Figure 2 . For the achievement measures, there was a moderate gap (> .5 z score) between the IQ-discrepant and low-achieving-only LD students in word problem solving but virtually no gap in most of the other measures with the exception of a small gap (< .5 z score) in math problem solving-nondefinition. The gap between no-LD and LD students was consistently large (approximately .75 or greater z score) across achievement measures. For the cognitive/behavioral measures, there was a moderate to large gap between the no-LD and LD groups across measures. The gap was generally small to nonexistent across measures between the IQ-discrepant and low-achieving-only LD students with the exception of working memory. However, this noticeable difference in working memory favoring IQ-discrepant LD students did not rise to the level of a statistically significant shape difference in the multivariate context.

Achievement (a) and cognitive (b) profiles of IQ-ACH definitional groups.

Note. IQ-ACH group = IQ-achievement-discrepant versus strictly low-achieving learning disability; mps-nd = math problem solving-nondefinition; wps = word problem solving; arith = arithmetic; comp = computational fluency; wi = word identification; sw = sight word fluency; read = word reading; pc = passage comprehension; viq = verbal IQ; nviq = nonverbal IQ; lang = language; cf = concept formation; ltm = long-term memory; wm = working memory; stm = short-term memory; ps = processing speed; att = attentive behavior.

Canonical Analyses

La-ld comparison groups (excluding no ld).

For the achievement measures, the canonical analysis results were consistent with the shape effects in the profile analyses. The magnitudes of the standardized discriminant function and correlation coefficients were generally greater among the math ( r s = .53 to .78; see Table 3 ) than reading ( r s = .22 to .48) outcomes and greater among word reading ( r = .39) and passage comprehension ( r = .48) than word identification and sight word efficiency ( r = .22 for both). The canonical correlation was highest for general arithmetic (.78 compared to the next highest, .59, for math problem solving-nondefinition; see Table 3 ), suggesting that general arithmetic is the most discriminating of the achievement measures between LA and LD students. For the cognitive/behavioral variables, the canonical correlations were generally low to moderate with the exception of concept formation ( r = .79; see Table 3 ), which appears to be the most discriminating of the cognitive measures between LA and LD students.

IQ-ACH comparison groups (excluding no LD)

For the achievement measures, the canonical analysis results were mostly consistent with the shape effects in the profile analyses. However, whereas the IQ-discrepant and low-achieving-only LD students differed only on word problem solving in the profile analysis, based on the canonical correlations, math problem solving-nondefinition, word problem solving, and computational fluency appear to be similarly discriminating between the two groups ( r s = .47, .48, and −.42, respectively; see Table 4 ). The magnitude of the canonical correlations for the reading measures and general arithmetic were substantially lower ( r s = −.15 to .24). For the cognitive/behavioral variables, the canonical correlations were generally low to moderate with the exception of language and working memory ( r = .69 and .68; see Table 4 ), which appears to be similarly discriminating between LA and LD students.

MANOVA and Regression Results (IQ-ACH Definitional Group Comparisons).

Note . IQ-ACH = IQ-achievement-discrepant versus strictly low-achieving learning disability; mps-nd = math problem solving-nondefinition; wps = word problem solving; arith = arithmetic; comp = computational fluency; wi = word identification; sw = sight word fluency; rd = word reading; pc = passage comprehension; NO LD = students do not meet criteria for low achieving or learning disabled; ACH = learning disability based on low achievement in math problem solving only; sdfc = standardized discriminant function coefficient; r = canonical correlation; lang = language; cf = concept formation; ltm = long-term memory; wm = working memory; stm = short-term memory; ps = processing speed; att = attentive behavior.

Regression Analyses

In addition to the canonical analysis, the regression analyses provide converging evidence that general arithmetic is the key discriminator between LA and LD students. It also discriminates between no LD and generally LA in math problem solving. When controlling for math problem solving-definition, arithmetic was the only achievement variable on which no-LD students differed statistically from all low-achieving students (no LD vs. LA/LD: t (809) = −2.20, p < .05) and LA students were statistically different from LD students (LA vs. LD: t (809) = −2.37, p < .05; see Table 3 for all other comparisons). The only other statistical differences when controlling for math problem solving-definition were between no LD and the low-achieving students (LA/LD) on sight word fluency, t (809) = −2.41, p < .05, and general reading, t (809) = −2.01, p < .05. When controlling for the group definition variable, math problem solving, there were no statistical differences between the no-LD and low-achieving students or between LA and LD students on any of the cognitive/behavioral measures (see Table 3 ).

The regression analyses results were consistent with both the profile and canonical analyses supporting word problem solving as the main discriminator between IQ-discrepant and low-achieving-only LD students. It was the only achievement measure for which the two groups differed statistically when controlling for both math problem solving-definition and IQ, t (809) = −2.31, p < .05. The only other statistical difference in achievement when controlling for math problem solving-definition and IQ was between no LD and LD (both IQ discrepant and low achieving only) on general arithmetic, t (809) = −2.04, p < .05 (see Table 4 for all other comparisons). For the cognitive/behavioral measures, the only statistical difference between groups when controlling for math problem solving-definition and IQ was between IQ-discrepant and low-achieving-only LD on working memory, t (809) = −2.01, p < .05 (see Table 4 for all other comparisons). This was consistent with the canonical analyses but inconsistent with the profile analyses, which suggest that when accounting for all cognitive/behavior variables at once, there is no statistical difference (i.e., shape effect) between the groups.

Our goal was to evaluate whether groups formed according to different definitions of math-problem-solving LD differ in achievement and cognitive/behavioral outcomes. We evaluated group differences according to two research traditions: (a) low achieving (< 25th percentile) versus LD (< 10th percentile) and (b) IQ-achievement-discrepant versus low-achieving-only LD. We based our evaluations on three statistical methods: profile and canonical analyses to control for the relations among the nondefinition variables and regression analyses to control for the relations between the definition and nondefinition variables.

The key finding was that regardless of how the groups are defined, there are shape differences in the profiles among the achievement but not the cognitive/behavioral measures. When the groups are defined strictly by math-problem-solving achievement level (LA vs. LD), they differ only in basic arithmetic skills. When the groups are defined by IQ discrepancy (IQ-discrepant vs. low-achieving LD only), they differ only in word problem solving. Working memory is the only cognitive/behavioral variable on which the IQ-discrepant and low-achieving-only students differ (even when controlling for the group definition variables). However, this difference does not appear to be distinct in the context of other cognitive/behavioral variables due to the lack of a shape effect between the groups.

Although there are moderate differences in other achievement and cognitive abilities between LD groups (regardless of how the groups are defined), most of these differences can be explained by the relations among the nondefinition measures or between the definition and non-definition measures. That is, math problem solving is correlated with most measures of achievement, cognitive abilities, and attentive behavior, and most of these measures are correlated with each other. Therefore, students who tend to do well in math problem solving tend to do well on all of these measures, and students who are poor math problem solvers tend to do poorly on all of these measures. This pattern is also true for IQ.

Even so, there are two measures in which there are greater differences in groups than would be expected given the relations among math problem solving, IQ, and the other measures of achievement, cognitive abilities, and attentive behavior. These measures are basic arithmetic and word problem solving.

For the LA-LD comparisons, the univariate differences across math measures and in concept formation and working memory are consistent with other studies based on this method of defining groups ( Geary et al., 2007 ; Murphy et al., 2007 ). Our study demonstrates that for most of these measures with the exception of general arithmetic, the differences between the groups are explained by the difference in their problem-solving ability. However, it is not clear why basic arithmetic is unique among the other math skills in distinguishing these two groups. One possible explanation is that math problem solving-nondefinition and word problem solving are relatively similar to WJIII Applied Problems in that they emphasize language comprehension and math reasoning over computation. It is therefore not unexpected that when controlling for performance on Applied Problems , there is no difference between groups in the other problem-solving measures. The relatively low-level computational skills required for computational fluency may also be why level differences on Applied Problems (i.e., group definition) explain any differences in computational fluency. Arithmetic (i.e., the Wide Range Achievement Test, Arithmetic subtest) requires decontextualized counting, identifying numbers, number comparison, and written computations, which is a broader range of skills than those required for computational fluency but consistent with those of Applied Problems. It may be that the ability to work with numbers in a variety of ways in the absence of context is what uniquely (relative to all other achievement and cognitive measures) distinguishes different levels of problem-solving ability.

There are several numerical abilities related to third-grade math achievement that were not evaluated in this study, including counting knowledge ( Geary, 2004 ), estimation ( Jordan & Hanich, 2003 ), number line estimation ( Geary, Bailey, & Hoard, 2009 ; Sasanguie, Göbel, Moll, Smets, & Reynvoet, 2013 ), number sets ( Geary et al., 2009 ), digit comparison ( Sasanguie et al., 2013 ), and the approximate number system ( Halberda, Mazzocco, & Feigenson, 2008 ). These numerical abilities are sometimes grouped under the term “number sense.” It has been suggested that number sense is to math as phonological awareness is to reading and as such is an important indicator of future math performance ( Gersten & Chard, 1999 ). Nonsymbolic number line estimation and number sets may be especially relevant to third-grade math achievement ( Sasanguie et al., 2013 ). It is not clear if these number sense abilities are captured by the group differences in arithmetic or if they would uniquely explain (compared to arithmetic) problem-solving differences in the groups. It is possible that the inclusion of measures of nonsymbolic number line estimation and number sets in the multivariate analysis would explain the group differences in general arithmetic.

For the IQ-discrepant/low-achieving-only comparisons, there were univariate differences only in word problem solving and working memory, even when controlling for the group definition variables. In terms of the univariate results, the IQ-ACH group differences in working memory and word problem solving were not consistent with other evidence ( González & Espinel, 1999 , 2002 ). There are several differences between this study and the González and Espinel studies that may account for the different results. González and Espinel used an arithmetic measure to define LD groups, whereas we used math problem solving as the definition variable. Their working memory task was a dot span task requiring counting dots and remembering numbers, whereas ours was a listening recall task requiring sentence comprehension and remembering words. The former is a numerical working memory task, whereas the latter is a working memory task involving language. It should be noted that in González and Espinel’s study, the IQ-achievement-discrepant students were higher on vocabulary (i.e., comparable to differences in listening recall in our study) than the strictly low-achieving students. Also, in our study, there was no difference between the groups on the numbers reversed short-term memory task (i.e., comparable to the lack of difference in numerical working memory in the González and Espinel studies). Finally, the word-problem-solving tasks were very similar across studies, both involving orally presented story problems involving relatively simple computations.

The working memory measure used in this study (i.e., Listening Recall ) requires language comprehension as well as short-term memory for words. The IQ-discrepant and low-achieving-only LD group difference in working memory is not unique among the cognitive/behavioral measures given the lack of a shape effect. Yet it is possible the language and memory requirements shared across the cognitive measures but fully encapsulated in the working memory measure explains the group difference in word problem solving (which is unique among the math measures given the shape effect in achievement outcomes). That is, IQ-achievement-discrepant students who are very low achieving (10th percentile) have a greater than expected advantage in language and short-term memory for words (hence their greater IQ) over very-low-achieving students who are not IQ discrepant. This gives IQ-discrepant students an advantage over nondiscrepant students in a word-problem-solving task that requires greater oral language than computational resources (e.g., the word-problem-solving tasks with simple computations) but not on problem-solving tasks that are more demanding of computational resources (e.g., the math-problem-solving definition variable that is highly correlated with arithmetic). However, the advantage that IQ-discrepant students have over nondiscrepant students in word problem solving may not be functionally meaningful given both groups’ extremely low performance relative to non-LD students.

In comparisons of IQ-achievement-discrepant and low-achieving-only students, the results from this study are consistent with other literature focused on comparisons between children with MDs but no reading disabilities (MD) and those with both math and reading disabilities (MD/RD) ( Jordan & Hanich, 2003 ; Jordan, Hanich, & Kaplan, 2003b ; Jordan & Montani, 1997 ). The MD children from these studies may be somewhat comparable to the IQ-achievement-discrepant students (who do not differ from no-LD students on IQ), and the MD/RD children are somewhat comparable to the low-achieving-only students (who have lower IQs than both IQ-achievement-discrepant and no-LD students) based on average levels of IQ. Although MD and MD/RD children are equally poor at retrieving math facts under timed conditions, the MD children perform better than MD/RD children when there is no time constraint ( Jordan & Montani, 1997 ). In untimed conditions, MD children are able to use backup counting strategies to outperform MD/RD children. Math fact mastery (i.e., efficient retrieval from long-term memory) appears to be independent of language and reading abilities ( Jordan, Hanich, & Kaplan, 2003a ). MD children who have poor math fact mastery but average language and reading abilities may be able to leverage language abilities in applying backup strategies to perform simple calculations, although these language abilities do not compensate for nonverbal number sense abilities (e.g., estimation; Jordan & Hanich, 2003 ). In the case of number sense abilities, MD children do not differ from MD/RD. It the context of this study, MD children would be expected to outperform MD/RD children on untimed math tasks where language abilities could be used to compensate for poor math fact mastery and poor number sense abilities (similar to IQ vs. ACH students on word problem solving) but perform comparably to MD/RD children on arithmetical tasks that require timed retrieval of math facts or number sense abilities (similar to IQ vs. ACH students on computational fluency, general arithmetic, and applied problem solving).

Practical Implications

It is possible that the IQ-discrepant/-nondiscrepant distinction may warrant different types of intervention, perhaps focusing on working memory and related cognitive resources, for example, helping IQ-discrepant students identify and use problem-solving strategies that leverage their relative advantage (compared to low-achieving-only LD students) for processing and holding verbal information in short-term memory. Future studies establishing the effectiveness of intervening in this way are needed to evaluate the potential of investing resources into this type of individualized instruction. In making these distinctions, it should be recognized that there is overlap between the groups in working memory and it is best to directly assess working memory regardless of IQ discrepancy. A stronger argument could be made for focusing on the practical implications of arithmetic as a distinguishing characteristic between math-problem-solving-LD and non-LD students.

Students who are low achieving in math problem solving, whether they meet criteria for LD or not, appear to have a particular problem with general arithmetic, although it is not clear if the problem is with procedural computational abilities, number sense, or a combination of the two. Targeted intervention that focuses on improving these students’ general arithmetic including procedural skills and knowledge about relations between quantities (i.e., number sense) may be a necessary first step to improving math-problem-solving skills.

Limitations

A criticism of research evaluating cognitive factors related to LD is that we do not know if they are causal, or a consequence of the LD, or a covariate ( Büttner & Hasselhorn, 2011 ). This study does not address these possibilities. The results from this study should be treated as preliminary evidence for future studies that use other methods (e.g., experimental, longitudinal) that evaluate whether the potential markers of problem-solving LD identified by this study predict problem-solving performance or can be manipulated to improve performance.

Acknowledgments

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported in part by grants from the Eunice Kennedy Shriver National Institute of Child Health and Human Development (Grant RO1 HD46154 and Core Grant HD15052 to Vanderbilt University, and Grant K99HD061689 to the University of Houston). The content is solely the responsibility of the authors and does not necessarily represent the official views of the Eunice Kennedy Shriver National Institute of Child Health and Human Development or the National Institutes of Health.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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Common Deficits With Learning Disabilities in Math

• Applied Math

Behavioral Problems

• If Your Child Is Struggling

Identifying specific skill deficits is the first step educators take when they attempt to design appropriate instruction for a child diagnosed with learning disabilities in basic math or applied math.

Special education teachers typically use  standardized diagnostic assessment , observations, and analysis of student work to identify specific areas of weakness. Teachers then develop instruction and select appropriate strategies.

Are you concerned that your child might have a learning disability in math? Speak with your child's teacher, principal or school counselor if she has any of the signs of math learning disabilities covered in this review.

Learning Disabilities in Basic Math

Children with learning disabilities in math may have difficulty with remembering math facts, steps in problem-solving, complex rules, and formulas. They may struggle to understand the meaning of math facts, operations, and formulas.

Such children also tend to struggle to solve problems quickly and efficiently or focus attention on details and accuracy. They might have difficulty mentally computing answers and fail to understand math terms.

Learning Disabilities in Applied Math

Students with a learning disability in applied math, in particular, may fail to understand why problem-solving steps are needed and how rules and formulas affect numbers and the problem-solving process.

They may get lost in the problem-solving process and find themselves unable to apply math skills in new problem-solving situations.

Remembering and following multi-step instructions may prove especially challenging for these children. In some cases, they might make errors while problem-solving due to poor handwriting. They may also be unable to make logical leaps in problem-solving based on previous learning or experience the inability to find the important information in a word problem. Choosing the right problem-solving strategy to correctly solve word problems will stump these children as well.

While their peers may be able to to find errors in their own work or to identify mistakes they made in solving the problem, children with learning disabilities in applied math will find it impossible or painstaking to do so.

Parents and teachers may notice the student's struggles while assessing his work or may hear him directly mentioning such problems.

Some students with learning disabilities in math may act out to avoid doing math work.

If a normally well-behaved child is acting out in math class, a learning disability may be the cause.

Some students with disabilities won't act out but will avoid math class by feigning sickness or withdrawing from the teacher or their peers in class.

What to Do If Your Child Is Struggling

When you observe these problems in your child's work, share the information with his teachers to help develop appropriate instructional strategies that target your child's specific needs. You can also ask your child where he feels he struggles most with math and request that he be evaluated.

If you suspect that your child has a learning disability in either basic or applied math, consult a school faculty member at once. Remember that early intervention is key. Rather than ignoring the problem, it's best to address it immediately to prevent it from taking a toll on your child's grades and self-esteem.

By Ann Logsdon Ann Logsdon is a school psychologist specializing in helping parents and teachers support students with a range of educational and developmental disabilities. 

Math Learning Disabilities

While children with disorders in mathematics are specifically included under the definition of Learning Disabilities, seldom do math learning difficulties cause children to be referred for evaluation. In many school systems, special education services are provided almost exclusively on the basis of children’s reading disabilities. Even after being identified as learning disabled (LD), few children are provided substantive assessment and remediation of their arithmetic difficulties.

This relative neglect might lead parents and teachers to believe that arithmetic learning problems are not very common, or perhaps not very serious. However, approximately 6% of school-age children have significant math deficits and among students classified as learning disabled, arithmetic difficulties are as pervasive as reading problems. This does not mean that all reading disabilities are accompanied by arithmetic learning problems, but it does mean that math deficits are widespread and in need of equivalent attention and concern.

Evidence from learning disabled adults belies the social myth that it is okay to be rotten at math. The effects of math failure throughout years of schooling, coupled with math illiteracy in adult life, can seriously handicap both daily living and vocational prospects. In today’s world, mathematical knowledge, reasoning, and skills are no less important than reading ability .

Different types of math learning problems

As with students’ reading disabilities, when math difficulties are present, they range from mild to severe. There is also evidence that children manifest different types of disabilities in math. Unfortunately, research attempting to classify these has yet to be validated or widely accepted, so caution is required when considering descriptions of differing degrees of math disability. Still, it seems evident that students do experience not only differing intensities of math dilemmas, but also different types, which require diverse classroom emphases, adaptations and sometimes even divergent methods.

Mastering basic number facts

Many learning disabled students have persistent trouble “memorizing” basic number facts in all four operations, despite adequate understanding and great effort expended trying to do so. Instead of readily knowing that 5+7=12, or that 4x6=24, these children continue laboriously over years to count fingers, pencil marks or scribbled circles and seem unable to develop efficient memory strategies on their own.

For some, this represents their only notable math learning difficulty and, in such cases, it is crucial not to hold them back “until they know their facts.” Rather, they should be allowed to use a pocket-size facts chart in order to proceed to more complex computation, applications, and problem-solving. As the students demonstrate speed and reliability in knowing a number fact, it can be removed from a personal chart. Addition and multiplication charts also can be used for subtraction and division respectively. For specific use as a basic fact reference, a portable chart (back-pocket-size, for older students) is preferable to an electronic calculator. Having the full set of answers in view is valuable, as is finding the same answer in the same location each time since where something is can help in recalling what it is. Also, by blackening over each fact that has been mastered, overreliance on the chart is discouraged and motivation to learn another one is increased. For those students who have difficulty locating answers at the vertical/horizontal intersections, it helps to use cutout cardboard in a backward L-shape.

Several curriculum materials offer specific methods to help teach mastering of basic arithmetic facts. The important assumption behind these materials is that the concepts of quantities and operations are already firmly established in the student’s understanding. This means that the student can readily show and explain what a problem means using objects, pencil marks, etc. Suggestions from these teaching approaches include:

• Interactive and intensive practice with motivational materials such as games …attentiveness during practice is as crucial as time spent
• Distributed practice, meaning much practice in small doses …for example, two 15-minute sessions per day, rather than an hour session every other day
• Small numbers of facts per group to be mastered at one time …and then, frequent practice with mixed groups
• Emphasis is on “reverses,” or “turnarounds” (e.g., 4 + 5/5 + 4, 6x7/7x6) …In vertical. horizontal, and oral formats
• Student self-charting of progress …having students keep track of how many and which facts are mastered and how many more there are to go
• Instruction, not just practice …Teaching thinking strategies from one fact to another (e.g., doubles facts, 5 + 5, 6 + 6, etc. and then double-plus-one facts, 5 + 6, 6 + 7, etc.).

(For details of these thinking strategies, see Garnett, Frank & Fleischner, 1983, Thornton.1978; or Stern, 1987).

Arithmetic weakness/math talent

Some learning disabled students have an excellent grasp of math concepts, but are inconsistent in calculating. They are reliably unreliable at paying attention to the operational sign, at borrowing or carrying appropriately, and at sequencing the steps in complex operations. These same students also may experience difficulty mastering basic number facts.

Interestingly, some of the students with these difficulties may be remedial math students during the elementary years when computational accuracy is heavily stressed, but can go on to join honors classes in higher math where their conceptual prowess is called for. Clearly, these students should not be tracked into low level secondary math classes where they will only continue to demonstrate these careless errors and inconsistent computational skills while being denied access to higher-level math of which they are capable. Because there is much more to mathematics than right-answer reliable calculating, it is important to access the broad scope of math abilities and not judge intelligence or understanding by observing only weak lower level skills. Often a delicate balance must be struck in working with learning disabled math students which include:

• Acknowledging their computational weaknesses
• Maintaining persistent effort at strengthening inconsistent skills;
• Sharing a partnership with the student to develop self-monitoring systems and ingenious compensations; and at the same time, providing the full, enriched scope of math teaching.

The written symbol system and concrete materials

Many younger children who have difficulty with elementary math actually bring to school a strong foundation of informal math understanding. They encounter trouble in connecting this knowledge base to the more formal procedures, language, and symbolic notation system of school math. The collision of their informal skills with school math is like a tuneful, rhythmic child experiencing written music as something different from what he/she already can do. In fact, it is quite a complex feat to map the new world of written -math symbols onto the known world of quantities, actions and, at the same time to learn the peculiar language we use to talk about arithmetic. Students need many repeated experiences and many varieties of concrete materials to make these connections strong and stable. Teachers often compound difficulties at this stage of learning by asking students to match pictured groups with number sentences before they have had sufficient experience relating varieties of physical representations with the various ways we string together math symbols, and the different ways we refer to these things in words. The fact that concrete materials can be moved, held, and physically grouped and separated makes them much more vivid teaching tools than pictorial representations. Because pictures are semiabstract symbols, if introduced too early, they easily confuse the delicate connections being formed between existing concepts, the new language of math, and the formal world of written number problems.

In this same regard, it is important to remember that structured concrete materials are beneficial at the concept development stage for math topics at all grade levels. There is research evidence that students who use concrete materials actually develop more precise and more comprehensive mental representations, often show more motivation and on-task behavior, may better understand mathematical ideas, and may better apply these to life situations. Structured, concrete materials have been profitably used to develop concepts and to clarify early number relations, place value, computation, fractions, decimals, measurement, geometry, money, percentage, number bases story problems, probability and statistics), and even algebra.

Of course, different kinds of concrete materials are suited to different teaching purposes (see appendix for selected listing of materials and distributors). Materials do not teach by themselves; they work together with teacher guidance and student interactions, as well as with repeated demonstrations and explanations by both teachers and students.

Often students’ confusion about the conventions of written math notation are sustained by the practice of using workbooks and ditto pages filled with problems to be solved. In these formats, students learn to act as problem answerers rather than demonstrators of math ideas. Students who show particular difficulty ordering math symbols in the conventional vertical, horizontal, and multi-step algorithms need much experience translating from one form to another. For example, teachers can provide answered addition problems with a double box next to each for translating these into the two related subtraction problems. Teachers can also dictate problems (with or without answers) for students to translate into pictorial form, then vertical notation, then horizontal notation. It can be helpful to structure pages with boxes for each of these different forms.

Students also can work in pairs translating answered problems into two or more different ways to read them (e.g., 20 x 56 - 1120 can be read twenty times fifty- six equals one thousand, one hundred and twenty or twenty multiplied by fifty-six is one thousand, one hundred, twenty). Or, again in pairs, students can be provided with answered problems each on an individual card; they alternate in their demonstration, or proof, of each example using materials (e.g., bundled sticks for carrying problems). To add zest, some of the problems can be answered incorrectly and a goal can be to find the “bad eggs.”

Each of these suggestions is intended to move youngsters out of the rut of thinking of math as getting right answers or giving up. They help create a frame of mind that connects understanding with symbolic representation, while attaching the appropriate language variations.

The language of math

Some LD students are particularly hampered by the language aspects of math, resulting in confusion about terminology, difficulty following verbal explanations, and/or weak verbal skills for monitoring the steps of complex calculations. Teachers can help by slowing down the pace of their delivery, maintaining normal timing of phrases, and giving information in discrete segments. Such slowed down “chunking” of verbal information is important when asking questions, giving directions, presenting concepts, and offering explanations.

Equally important is frequently asking students to verbalize what they are doing. Too often, math time is filled either with teacher explanation or with silent written practice. Students with language confusions need to demonstrate with concrete materials and explain what they are doing at all ages and all levels of math work, not just in the earliest grades. Having students regularly “play teacher” can be not only enjoyable but also necessary for learning the complexities of the language of math. Also, understanding for all children tends to be more complete when they are required to explain, elaborate, or defend their position to others; the burden of having to explain often acts as the extra push needed to connect and integrate their knowledge in crucial ways.

Typically, children with language deficits react to math problems on the page as signals to do something, rather than as meaningful sentences that need to be read for understanding. It is almost as though they specifically avoid verbalizing. Both younger and older students need to develop the habit of reading or saying problems before and/or after computing them. By attending to the simple steps of self-verbalizing, they can monitor more of their attentional slips and careless errors. Therefore, teachers should encourage these students to:

• Stop after each answer,
• Read aloud the problem and the answer, and
• Listen to myself and ask, “Does that make sense?”

For youngsters with language weakness, this may take repeated teacher modeling, patient reminding and much practice using a cue card as a visual reminder.

Visual-spatial aspects of math

A small number of LD students have disturbances in visual-spatial-motor organization, which may result in weak or lacking understanding of concepts, very poor “number sense,” specific difficulty with pictorial representations and/or poorly controlled handwriting and confused arrangements of numerals and signs on the page. Students with profoundly impaired conceptual understanding often have substantial perceptual-motor deficits and are presumed to have right hemisphere dysfunction.

This small subgroup may well require a very heavy emphasis on precise and clear verbal descriptions. They seem to benefit from substituting verbal constructions for the intuitive/spatial/relational understanding they lack. Pictorial examples or diagrammatic explanations can thoroughly confuse them, so these should not be used when trying to teach or clarify concepts. In fact, this subgroup is specifically in need of remediation in the area of picture interpretation, diagram and graph reading, and nonverbal social cues. To develop an understanding of math concepts, it may be useful to make repeated use of concrete teaching materials (e.g., Stern blocks, Cuisenaire rods), with conscientious attention to developing stable verbal renditions of each quantity (e.g., 5), relationship (e.g., 5 is less than 7), and action (e.g., 5+2=7). Since understanding visual relationships and organization is difficult for these students, it is important to anchor verbal constructions in repeated experiences with structured materials that can be felt, seen, and moved around as they are talked about. For example, they may be better able to learn to identify triangles by holding a triangular block and saying to themselves, “A triangle has three sides. When we draw it, it has three connected lines.” For example, a college freshman who had this deficit could not “see” what a triangle was without saying this to herself when she looked at different figures or attempted to draw a triangle.

The goal for these students is to construct a strong verbal model for quantities and their relationships in place of the visual-spatial mental representation that most people develop. Consistent descriptive verbalizations also need to become firmly established in regard to when to apply math procedures and how to carry out the steps of written computation. Great patience and verbal repetition are required to make small incremental steps.

It is important to recognize that average, bright, and even very bright youngsters can have the severe visual-spatial organization deficits that make developing simple math concepts extremely difficult. When such deficits are accompanied by strong verbal skills, there is a tendency to disbelieve the impaired area of functioning. Thus, parents and teachers can spend years growling, “She’s just not trying…She doesn’t play attention…She must have a math phobia…It’s probably an emotional problem.” Because other accompanying weaknesses usually include a poor sense of body in space, difficulty reading the nonverbal social signals of gesture and face, and often nightmarish disorganization in the world of “things,” it can be easy to mistake the problem for a constellation of emotional symptoms. Misreading the problems in this way delays the appropriate work that is needed both in mathematics and the other areas.

Math learning difficulties are common, significant, and worthy of serious instructional attention in both regular and special education classes. Students may respond to repeated failure with withdrawal of effort, lowered self-esteem, and avoidance behaviors. In addition, significant math deficits can have serious consequences on the management of everyday life as well as on job prospects and promotion.

Math learning problems range from mild to severe and manifest themselves in a variety of ways. Most common are difficulties with efficient recall of basic arithmetic facts and reliability in written computation. When these problems are accompanied by a strong conceptual grasp of mathematical and spatial relations, it is important not to bog the student down by focusing only on remediating computation. While important to work on, such efforts should not deny a full math education to otherwise capable students.

Language disabilities, even subtle ones, can interfere with math learning. In particular, many LD students have a tendency to avoid verbalizing in math activities, a tendency often exacerbated by the way math is typically taught in America. Developing their habits of verbalizing math examples and procedures can greatly help in removing obstacles to success in mainstream math settings.

Many children experience difficulty bridging informal math knowledge to formal school math. To build these connections takes time, experiences, and carefully guided instruction. The use of structured, concrete materials is important to securing these links, not only in the early elementary grades, but also during concept development stages of higher-level math. Some students need particular emphasis on the translating between different written forms, different ways of reading these, and various representations (with objects or drawings) of what they mean.

An extremely handicapping, though less common math disability, derives from significant visual-spatial-motor disorganization. The formation of foundation math concepts is impaired in this small subgroup of students. Methods to compensate include avoiding the use of pictures or graphics for conveying concepts, constructing verbal versions of math ideas, and using concrete materials as anchors. The organizational and social problems that accompany this math disability are also in need of long-term appropriate remedial attention in order to support successful life adjustment in adulthood.

In sum, as special educators, there is much we can and need to do in this area that calls for so much greater attention than we have typically provided.

About the author

Dr. Garnett received her doctorate from Teachers College, Columbia University. Over the last 18 years Dr. Garnett has been on the faculty of the Department of Special Education, Hunter College, CUNY where she directs the masters program in Learning Disorders. She is currently with The Edison Project, where she is the architect of their Responsible Inclusion/Special Edison Support.

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Disabilities in math affect many students — but get little attention

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Laura Jackson became seriously concerned about her daughter and math when the girl was in third grade. While many of her classmates flew through multiplication tests, Jackson’s daughter struggled to complete her 1 times table. She relied on her fingers to count, had difficulty reading clocks and frequently burst into tears when asked at home to practice math flashcards. At school, the 9-year-old had been receiving help from a math specialist for two years, with little improvement. “We hit a point where she was asking me, ‘Mom, am I stupid?’” Jackson recalled. 

Then, when Jackson was having lunch with a friend one day, she heard for the first time about a disorder known as dyscalculia. After lunch, she went to her computer, looked up the term, and quickly came across a description of the learning disability, which impacts a child’s ability to process numbers, retain math knowledge and complete math problems. “I was like, ‘Oh my gosh, this is my kid,’” Jackson said.

Nationwide, hundreds of thousands of students face challenges learning math due to math disabilities like dyscalculia, a neurodevelopmental learning disorder caused by differences in the parts of the brain that are involved with numbers and calculations . There are often obstacles to getting help.

America’s schools have long struggled to identify and support students with learning disabilities of all kinds: Kids often languish while waiting to receive a diagnosis; families frequently have to turn to private, often pricey, providers to get one; and even with a diagnosis, some children still don’t get the supports they need because their schools are unable to provide them.

That’s slowly changing — for some disabilities. A majority of states have passed laws that mandate screening early elementary students for the most common reading disability, dyslexia, and countless districts train teachers how to recognize and teach struggling readers. Meanwhile, parents and experts say school districts continue to neglect students with math disabilities like dyscalculia, which affects up to 7 percent of the population and often coexists with dyslexia.

“Nobody uses the proper term for it, it’s not diagnosed frequently,” said Sandra Elliott, a former special education teacher and current chief academic officer at TouchMath, a multisensory math program. “We’re all focused on literacy.”

The Math Problem  

Sluggish growth in math scores for U.S. students began long before the pandemic, but the problem has snowballed into an education crisis. This back-to-school season, the Education Reporting Collaborative, a coalition of eight newsrooms, will be documenting the enormous challenge facing our schools and highlighting examples of progress. The three-year-old Reporting Collaborative includes AL.com, The Associated Press, The Christian Science Monitor, The Dallas Morning News, The Hechinger Report, Idaho Education News, The Post and Courier in South Carolina, and The Seattle Times.

AI might disrupt math and computer science classes – in a good way

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Nationwide, teachers report that up to 40 percent of their students perform below grade level in math. And while students with math disabilities may be especially far behind , math scores for all students have remained dismal for years , showing that more attention needs to be paid to math instruction. Experts say learning the most effective methods for teaching students with math disabilities could significantly strengthen math instruction for all students. “You’ve got a huge number of students that are in the middle ground,” when it comes to math achievement but may not have a disability, Elliott added. Those students could also be helped by having explicit, multisensory instruction in math. “If it works for the students with the most severe disconnections and slower processing speeds, it’s still going to work for the kids that are in the ‘middle’ with math difficulties.”

“It’s not the fault of schools. I think it has to do with the amount of resources schools have to provide intervention to children, and reading takes priority over math.” Lynn Fuchs, research professor at Vanderbilt University

Covid exacerbated the nation’s problem with math achievement. The number of children who are several years behind in math has increased over the past few years and achievement gaps have widened. For some students, learning struggles may be due to an underlying disability like dyscalculia or other math learning disabilities that affect math calculation or problem solving skills. Yet only 15 percent of teachers report that their students have been screened for dyscalculia.

“There’s not as much research on math disorders or dyscalculia,” as there is on reading disabilities, said Karen Wilson, a clinical neuropsychologist who works with the organization Understood.org and specializes in the assessment of children with learning differences. “That also trickles down into schools.”

Related: Why it matters that Americans are comparatively bad at math

There are a host of reasons why math disabilities receive less attention than reading disabilities. Elementary teachers report more anxiety when it comes to teaching math, which can make it harder to teach struggling learners. Advocacy focused on math disabilities has been less widespread than that for reading disabilities. There is also a deep-seated societal belief that some people have a natural aptitude for math. “A lot of times, [parents] let it go for a long time because it’s culturally acceptable to be bad at math,” said Heather Brand, a math specialist and operations manager for the tutoring organization Made for Math.  

Some signs of dyscalculia are obvious at an early age, if parents and educators know what to look for. In the earliest years , a child might have difficulty recognizing numbers or patterns. Children may also struggle to connect a number’s symbol with what it represents, like knowing the number 3 corresponds to three blocks, for example. In elementary school, students may have trouble with math functions like addition and subtraction, word problems, counting money, or remembering directions.

Still, schools may be resistant to assessing math disabilities, or unaware of their prevalence. Even after Jackson learned about dyscalculia on her own, her daughter’s Seattle-area public school was doubtful that the third grader had a learning disability because she was performing so well in all other areas. Teachers suggested Jackson spend extra time on math at home. “For so many parents, they assume the school would let them know there’s an issue, but that’s just not how it works,” said Jackson. (She ultimately wrote a book, “Discovering Dyscalculia” about her family’s journey, and now runs workshops for parents of children with dyscalculia.)

Experts say universal screening, like those provided in many states for dyslexia, should be in place for math disabilities. Early diagnosis is crucial to provide children a stronger foundation in the early concepts that all math builds on. “Many times, if a student is caught early with the interventions that we all know work … these children can perform math, if not equal to their typically developing peers, they can get very, very close,” said Elliott from TouchMath.

Solving the Math Problem: Helping kids find joy and success in math

The Education Reporting Collaborative will host “Solving the Math Problem: Helping kids find joy and success in math,” a live expert panel, on Tuesday, Oct. 17 at 8 p.m. Eastern, 7 p.m. Central, 5 p.m. Pacific. This webinar is designed for families seeking strategies to help kids engage and excel in math.

Panelists include:Melissa Hosten , a Mathematics Outreach Co-Director at the University of Arizona, in the Department of Mathematics at the Center for Recruitment and Retention of Mathematics Teachers.

Elham Kazemi , a professor of mathematics education in the College of Education at the University of Washington.

The event registration shortlink is:  https://st.news/mathwebinar

As with other learning disabilities, a diagnosis is only the first step to getting children the help they need in school. In particular, students with dyscalculia often need a more structured approach to learning math that, like reading, involves “systematic and explicit” instruction and provides ample time to practice counting and recognizing numbers, said Lynn Fuchs, a research professor in special education and human development at Vanderbilt University. These students also may need strategies to help them commit math facts to memory, she added. To do this well, they often need small-group or one-on-one teaching, which is non-existent in many schools’ math instruction. “It’s not the fault of schools. I think it has to do with the amount of resources schools have to provide intervention to children, and reading takes priority over math,” said Fuchs.

Part of the problem is that teachers don’t receive the training needed to work with children with math disabilities. Teacher training programs offer little instruction on disabilities of any kind, and even less on math. In a 2023 survey by Education Week, nearly 75 percent of teachers reported that they had received little to no preservice or in-service training on supporting students with math disabilities. At least one state, Virginia, requires dyslexia awareness training for teacher licensure renewal, but has no similar requirement for math disability training. “It’s pretty rare for undergraduate degrees or even master’s degrees to focus on math learning disabilities with any level of breadth, depth, quality or rigor,” said Amelia Malone, director of research and innovation at the National Center for Learning Disabilities.

Nearly 75 percent of teachers reported in a 2023 Education Week survey that they had received little to no preservice or in-service training on supporting students with math disabilities..

Without more widespread knowledge of and support for dyscalculia, many parents have had to look for specialists and tutors on their own, which they say can be particularly challenging for math, and costly. Even after her daughter received a diagnosis, Jackson felt the girl’s school wasn’t supporting her enough. At school, her daughter’s math teacher demanded “tidy” math notebooks and discouraged drawing or doodling, activities that often helped the girl work through problems. In 2019, Jackson started pulling her daughter out of school for part of each day to teach her math at home. “I am not a math teacher, but I was so desperate,” Jackson said. “There’s no one who knows anything and we have to figure this out.”

Jackson pored over materials online and reached out to math disability experts in America and abroad for help. She started infusing her daughter’s math lessons with games and brought out physical objects, like small wooden rods, to help her practice counting. She worked with her daughter on the core foundations of math, including number sense and basic operations, to help establish the solid grounding that the girl was missing.

Experts say it’s possible to improve math outcomes for those who struggle, if more attention and resources are poured into math in the early years to ensure children do not reach third grade — or beyond — without the support they need.

Yet early childhood teachers are often the least equipped to teach math, especially for children with dyscalculia, said Marilyn Zecher, a dyslexia specialist who created a multisensory approach to math based on the popular Orton-Gillingham approach in reading. Zecher offers training on dyscalculia-related teaching strategies for teachers of all grade levels. Many of her strategies for early educators emphasize that math instruction starts through language. Children learn the basics of mathematics when teachers give them opportunities to verbally compare similarities and differences between objects, and describe how items or activities occur in relation to each other, such as “before” or “after.”

“The early ed teachers are the giants upon whose shoulders everybody else stands,” Zecher said. Early educators, like preschool teachers, not only teach foundational skills, they are also “so critical to identifying children who are having difficulties.”

Related: For teachers who fear math, banishing bad memories can help

At Brand’s organization, Made for Math , intensive tutoring based on Zecher’s approach often stands in for a lack of school-based support. Teachers create individualized lesson plans for students during each tutoring session, employing a variety of items to help students better understand math concepts. Students might use craft sticks bundled together to learn place value, cubes to learn subtraction or addition, and items that can be physically cut apart, like foam stickers, to learn fractions. Math specialists at the organization have found that children with dyscalculia need repetition, especially to understand math facts. Some students attend tutoring up to four days a week, at a cost of up to $1,000 a month . “It’s hard because it’s not something schools are offering, and kids deserve it,” said Brand.

In recent years, a handful of states, including Alabama , West Virginia and Florida , have introduced legislation that would require schools to identify and support younger students who struggle with math. Elliott’s company, TouchMath, introduced a screener earlier this year that can identify signs of math disabilities, like dyscalculia, in children as young as age 3.

“Many times, if a student is caught early with the interventions that we all know work…these children can perform math if not equal to their typically developing peers,” Sandra Elliott, a former special education teacher and current chief academic officer at TouchMath

Malone, from the National Center for Learning Disabilities, said, there are pockets of progress around the country in screening more children for math disabilities, but movement at the federal level — and in most states — is “nonexistent.”

New York City is one district that has prioritized math disability screening and math instruction in the early years. In 2015 and 2016, the city spent $6 million to roll out a new math curriculum featuring games, building blocks, art projects and songs. The district has also introduced universal math and reading screeners to try to identify students who may be behind.

Experts say that there are ways that all schools can make math instruction more accessible. In elementary schools, activities that involve more senses should be used more widely, including whole-body motions and songs for teaching numbers and hands-on materials for math operations. All students, and not only those with dyscalculia, could benefit from using manipulatives to help visualize problems and graph paper to assist in lining up numbers.  

As with dyslexia, figuring out better ways to teach kids with math disabilities will shore up math instruction across the board – and better meet students where they are. “Some kids won’t use [the strategies],” said Wilson, the neuropsychologist. “It’s really about having the option, so the student who’s struggling will be able to find a method that works for them.”

Jackson said her daughter could have benefited from a wider variety of methods at school. After several years of learning math at home, she was ready to try to re-join grade-level math classes. When the teen returned to school-based math classes in high school, she achieved an A in Algebra. “When you really understand what it is to be dyscalculic, then you can look around and decide what this person needs to succeed,” Jackson said. “It’s not just that you’re ‘bad at math’ and need to buckle down and try harder.”

This story about  dyscalculia  was produced by  The Hechinger Report , a nonprofit, independent news organization focused on inequality and innovation in education, as part of The Math Problem, an ongoing series about math instruction. The series is a collaboration with the Education Reporting Collaborative, a coalition of eight newsrooms that includes AL.com, The Associated Press, The Christian Science Monitor, The Dallas Morning News, The Hechinger Report, Idaho Education News, The Post and Courier in South Carolina, and The Seattle Times. Sign up for the Hechinger newsletter .

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Improving Math Problem Solving for Students with Learning Disabilities: Solve It!

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Problem solving is prominent in the current math curriculum reflecting national standards, yet solving mathematical word problems is a complex procedure involving multiple cognitive processes beyond computing an equation. Students must comprehend the problem information, create a mental model of the problem, and from that model determine and execute a viable solution plan. Particularly for students with learning disabilities who struggle in math, problem solving is daunting because of its variability: no series of discrete, procedural steps exist to consistently yield accurate solutions. This webinar gives you a research-based cognitive strategy instructional intervention,  Solve It! , including its theoretical foundations, core components, and application to current standards. Modifications for use with students with more severe disabilities, students who are English language learners, and younger students are also discussed.

This program is intended for fifth through eighth grade teachers of students with learning disabilities or those who struggle in math. Teachers should be familiar with grade level Common Core Standards in Mathematics as well as the eight Standards for Mathematical Practice. The program is appropriate for both general education math teachers and special education teachers who support students in inclusive settings.

After viewing this program, you will be able to:

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• Assess students’ problem-solving proficiency and differentiate instruction to reflect unique student needs.
• Incorporate weekly problem-solving instruction into mathematics pacing guides.

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How to ensure students with disabilities have an equal chance to succeed?

Second in a four-part series on non-apparent disabilities.

Nearly one in four Americans has a disability, according to the CDC . Many disabilities — such as chronic pain, learning disabilities, diabetes, and depression — are “non-apparent,” or not immediately obvious to others, presenting their own set of challenges.

So how can professors design their classes to give all students an equal chance to succeed?

The Law School’s Michael Ashley Stein — who has taught classes such as “Disability, Human Rights, and Development” at Harvard Law School and “Disability Law and Policy” at Harvard Kennedy School — finds it useful to draw connections across affinity groups.

“People of color might point out racism and women and others might point out sexism. To me, all those prejudices and civil- and human-rights-violating type actions are grouped under ableism,” said the visiting professor and co-founder and executive director at the Law School’s Project on Disability . “It’s important when teaching disability to create those linkages, and to create the kind of affinity and solidarity that reaches across groups.”

A tool favored by Andrew Clark , senior lecturer on music and director of choral activities at Harvard, is the Universal Design for Learning framework geared to different types of learners. It gives students options to demonstrate what they’ve learned.

“I have in the last 10 years been in many settings with students where they’ve talked openly about depression or anxiety, whereas 40 years ago, they would not have talked about it.” Arthur Kleinman

Clark began teaching “Music and Disability” in the Faculty of Arts and Sciences in 2016. He was inspired by his early years working at a music camp for individuals with disability and chronic illness and a desire to learn more about the intersection between music and disability studies. Students examine musicians or specific works with disability narratives and consider “how disability justice is enacted and embodied.”

“There’s a difference between accommodation and anticipation,” Clark said. “If we can design our classes — as well as extracurricular activities and student life — to anticipate every person rather than to accommodate everyone, that’s true inclusion. That’s making students feel empowered rather than accommodated.”

Nadine Gaab , associate professor at the Harvard Graduate School of Education, who has centered her work on non-apparent disabilities for more than 20 years, uses the messaging platform Slack to offer students multiple ways to communicate, incorporates scholars with disabilities in her syllabus, and has flexible participation and assignment submission policies.

This spring, Gaab is teaching “Children with Learning and Developmental Differences.” Students learn about the challenges faced by children with conditions such as dyslexia , dyscalculia, and dysgraphia, as well as the teachers, administrators, and medical teams working with them.

“We identify a number of different barriers that prevent us from delivering optimal care for children with invisible and visible disabilities,” Gaab said. “We then identify solutions in educational and community settings that could work in response to those challenges.”

Students apply what they’ve learned to real-world community spaces by identifying accessibility problems and proposing solutions — some of which have already been implemented, Gaab noted.

Mental health, particularly among students, is of interest to psychiatrist Arthur Kleinman . The Esther and Sidney Rabb Professor of Anthropology in the FAS, professor of medical anthropology in global health and social medicine, and professor of psychiatry at Harvard Medical School has been teaching for more than 40 years and has noticed marked changes in how students approach mental health issues.

“People are much more open about this,” said Kleinman, who noted that he used to discourage students from disclosing mental health issues because of stigma.

“I have in the last 10 years been in many settings with students where they’ve talked openly about depression or anxiety, whereas 40 years ago, they would not have talked about it. The responses they get today are also very different. They’re very supportive.”

While the professors say the stigma around non-apparent disabilities and mental health are lessening, they also agreed that more can be done to create more inclusive learning and research environments — at Harvard and beyond.

“It’s an extra cognitive load,” explained Gaab. “It’s extra-hard work to fit into a system that’s designed for the average learner. It’s really important to make sure that students recognize that in themselves, and faculty are aware, that it’s not that students are lazy or even ‘stupid.’ They’re really trying hard.”

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• University Disability Resources serves as the central resource for disability-related information, procedures, and services for the Harvard community.
• Students who wish to request accommodations should contact their School’s Local Disability Coordinator.
• The 24/7 mental health support line for students is 617-495-2042. Deaf or hard-of-hearing students can dial 711 to reach a Telecommunications Relay Service in their local area.
• Harvard Law School Project on Disability

Also in this series

Navigating Harvard with a non-apparent disability

4 students with conditions ranging from diabetes to narcolepsy describe daily challenges that may not be obvious to their classmates and professors

Why AI fairness conversations must include disabled people

Tech offers promise to help yet too often perpetuates ableism, say researchers. It doesn’t have to be this way.

Getting ahead of dyslexia

Harvard lab’s research suggests at-risk kids can be identified before they ever struggle in school

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ORIGINAL RESEARCH article

This article is part of the research topic.

Research on Teaching Strategies and Skills in Different Educational Stages

BIOTECHNOLOGY PROJECT-BASED LEARNING ENCOURAGES LEARNING AND MATHEMATICS APPLICATION Provisionally Accepted

• 1 Universidad Técnica Particular de Loja, Ecuador

The final, formatted version of the article will be published soon.

Project-based learning (PBL) is a promising approach to enhance mathematics learning concepts in higher education. Here, teachers provide guidance and support to PBL implementation. The objective of this study was to develop PBL-based biotechnological projects as a strategy for mathematics learning. The methodology design was applied to 111 university students from Biochemical, Chemical Engineering and Business Administration careers. Knowledge, skills, perceptions, and engagement were measured through questionnaires, workshops, rubrics, and survey instruments. As a result, the paired comparison between tests, questionnaires and project shows significant differences (P<.001) between the experimental group and the control group. It is concluded that the teaching of mathematics should be oriented to the development of competencies, abilities, and skills that allow students to generate real solutions and broaden their vision of the applicability of their knowledge using new learning strategies. Key words: Mathematical models, Biotechnology, Project based learning.Science, technology, engineering, and mathematics (STEM) education has become a crucial topic both inside and outside of school (Han et al., 2015). Currently, mathematics learning tends to be oriented towards textbooks, and students can only work on math problems based on what the teacher exemplifies; however, if given different contextbased problems, they will have difficulty solving them (Fisher et al., 2020). Likewise, the traditional classroom model does not encourage student's interest in STEM (Sahin, 2009) The research gap is between what students learn at the university and what they really need in the workplace (Holmes et al., 2015). Higher education institutions have been trying to provide students with both (i) hard skills, such as cognitive knowledge and professional skills (Vogler et al., 2018), and (ii) soft skills, such as problem-solving and teamwork (Lennox and Roos 2017). However, these skills are difficult to achieve through traditional learning. One learning that creates an active, collaborative atmosphere, and can increase selfconfidence in students is Project-based learning (PBL) (Cruz et al., 2022; Guo et al., 2022;Markula and Aksela 2022). The PBL method is applied as a teaching model that involves

Keywords: Mathematical Models, Biotechnology, Project Based Learning (PBL), Learning, Learning mathematics activities

Received: 02 Jan 2024; Accepted: 03 Apr 2024.

Copyright: © 2024 Vivanco and Jiménez-Gaona. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Mr. Oscar A. Vivanco, Universidad Técnica Particular de Loja, Loja, Ecuador

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