1. Demystifying Dyscalculia
Rochester Schools enrichment

2. DSM-V Criteria
Specific learning disorder with impairment in mathematics includes possible deficits in:
Number sense
Memorization of arithmetic facts
Accurate or fluent calculation
Accurate math reasoning
Mild: Some difficulties with learning in one or two academic areas, but may be able to compensate.
Moderate: Significant difficulties with learning, requiring some specialized teaching and some accommodations or supportive services
Severe: Severe difficulties with learning, affecting several academic areas and requiring ongoing intensive specialized teaching
Must be present for 6 months despite targeted help.

3. SLD in Mathematics Diagnosis
The diagnosis is based on a variety of methods, including:
Medical history,
Clinical interview
School report
Teacher evaluation
Rating scales
Psychometric tests
The IQ discrepancy criterion was abandoned, though that of age or class discrepancy criterion was retained. The application of a discrepancy is recommended by 1 to 2.5 SD.
All three specific developmental disorders are common (prevalence 5 %-15 %), occur early during the first years of formal schooling, and persist into adulthood.

4. ICD-10 Diagnostic criteria
Involves a specific impairment in arithmetical skills that tis not solely explicable on the basis of general mental retardation or of inadequate schooling. The deficit concerns mastery of basic calculation skills of addition, subtraction, multiplication, and division rather than of the more abstract mathematical skills involved in algebra, trigonometry, geometry, or calculus. Excludes arithmetic difficulties associated with a reading or spelling disorder and inadequate teaching.

5. No Specific standards for diagnosis
LDs in mathematics
Developmental arithmetic disorder
Developmental dyscalculia
Mathematics disabilities
Specific mathematical disabilities
Mathematics learning disability synonymously with dyslcaclulia
Specific usually implies that oral language, reading and writing are intact
Math deficits are frequently associated with other learning disabilities.

6. CORRELATES OF DEVELOPMENTAL DYSCALCULIA
Many children have and adolescents with dyslexia have associated cognitive dysfunction:
Impairment in working memory
Impairment in visuospatial skills
20% - 60% have comorbid disorders such as dyslexia and attention deficit disorder
Aversion to counting and arithmetic which turns into anxiety or generalized school phobia
Can impair personality development, schooling and occupational training
Adults with dyscalculia suffer a disadvantage in the job market

7. Development of numerical & arithmetic skills
Object Tracking System (OTS)
Allow rapid enumeration of small sets (4 or less)
ANS: Approximate Number System
Infants can can clearly distinguish one item from two items, and they are capable of a rough estimation of number for three or more items
Supports the estimation of the magnitude of a group without relying on language or symbols.
Precision of the ANS improves throughout childhood development and reaches a final adult level of approximately 15% accuracy, meaning an adult could distinguish 100 items versus 115 items without counting
Plays a crucial role in development of other numerical abilities, such as the concept of exact number and simple arithmetic
 Precision level of a child's ANS has been shown to predict subsequent mathematical achievement in school
Linked to the intraparietal sulcus of the brain

8. Development of arithmetic & numerical skills
ANS underlies the spatial-numerical association response code (SNARC) effect.
Tendency of larger numbers to be responded to faster by the right hand and lower numbers by the left hand, suggesting that the magnitude of a number is linked to a spatial representation.
Dehaene and other researchers believe this effect is caused by the presence of a “mental number line” in which small numbers appear on the left and increase as you move right. The SNARC effect indicates that the ANS works more effectively and accurately if the larger set of objects is on the right and the smaller on the left.

9. Development of arithmetic & numerical skills
Damage done to parietal lobe, specifically in the left hemisphere, can produce difficulties in counting and other simple arithmetic.
Damage directly to the intraparietal sulcus has been shown to cause acalculia, a severe disorder in mathematical cognition. Symptoms vary based the location of damage, but can include the inability to perform simple calculations or to decide that one number is larger than another.
Gerstmann syndrome a disease resulting in lesions in the left parietal and temoral , results in acalculia symptoms and further confirms the importance of the parietal region in the ANS.

10. Development of numerical & arithmetic skills
Recent research in developmental psychology and neuroscience has revealed a “ratio processing system” (RPS) sensitive to magnitudes of nonsymbolic ratios that may be ideally suited for grounding fractions concepts.
Individual differences in RPS acuity predict performance on four measures of mathematical competence, including a university algebra entrance exam.
Even abstract mathematical concepts like fractions may be grounded as much in basic nonsymbolic processing abilities as they are in higher-order logic and language.

11. Development of numerical & arithmetic skills
Participants completed nonsymbolic ratio comparisons across formats (panel a) more quickly than they
completed symbolic fraction comparisons (panel b), suggesting that nonsymbolic ratio comparisons were
made without conversion to symbolic form.

12. Development of numerical & arithmetic skills
Symbolic memory:
When language appears children become able to symbolize numbers linquistically with number-words
One to one correspondence in counting
Verbal arithmetic manipulation of quantities and numbers
Arabic place value numeral system
compared to one million seven hundred sixty-eight thousand three hundred twenty-nine – 62 letters long, not counting dashes and spaces.
Numerospatial conceptual ability (a “mental number line”) enabling operations with numerical symbols

13. ACADEMIC SKILL DEFICITS
Mathematics Calculations and Math Problem Solving Skills
Primary grades focus on whole numbers – understanding of numbers, calculations, and word problems.
Intermediate and middle school focus is on rational numbers – common fractions, decimals and proportions.
Subdomains include part-whole understanding, measurement interpretation, calculations and word problems.
High school – algebra, geometry, trigonometry, and calculus.
Latter are not considered domains of LDs as exemplified by the ICD-10 definition of specific arithmetical difficulties
Little is understood about how curricular components relate to one another, which aspects of mathematics performance are shared or distinct, how difficulty in one corresponds to difficulty in another, both concurrently and from one grade to the next, and whether instruction in one or another area produces better learning in a third domain.

14. Cognitive abilities associated with math skills
Math calculation skills are impacted by:
1. Controlled attention
2. Vocabulary knowledge
3. Visuospatial working memory
4 Phonoloical decoding and processing speed
Word problem skills are impacted by:
1. Teacher ratings of inattentive behavior
2. Nonverbal problem solving
3. Concept formation.
4. Language processing

15. Number-related core deficits
Representation of magnitude – i.e., automatic number sense:
Little evidence for support of diagnostic use of nonsymbolic magnitude compairon taks for assessments of children with or at risk for LDs in mathematis or for specific training in non-symbolic magnitude comparisons per se for these children at the present time.
Mapping symbols to magnitude:
Some evidence that children with intact approximate number systems, but are slower in mapping those quantities to the number words and Arabic numerals used in the formal mathematics system. Symbolic magnitude processing is more strongly related to math achievement than is nonsymbolic magnitude processing.

16. Broader view of math skill development
Mathematics comprises subdomains of knowledge, built on other general cognitive or neuropsychological systems:
1. Language system
2. Visual-spatial system
3. Executive control systems that sustains attention and inhibits irrelevant information
4. Working memory

17. Mathematics skills development
Calculations
Retrieval of facts (math fluency)
Application of procedural knowledge
Word problem solving involves calculations, language, reasoning, and reading skill.
Attention, organization, ability to switch sets, and work quickly enough to avoid overloading working memory stores.
Many math achievement tests tend to confound the multiple components of mathematics.

18. Core cognitive processes in mathematics
Comorbidity with reading disabilities (dyslexia):
Attentive behavior
Reasoning
Visuospatial memory
Rapid automatic naming

19. Cognitive processes & mathematics learning
Working memory
Impact both math calculations and word problems.
Attention
Ability to maintain attention across tasks that have multiple components and the unfold across time impact the ability to learn new math concepts and procecures and to performance on many math tasks.
Ability to flexibly switch between rules during task performance by focusing on task relevant information and inhibiting task irrelevant information. Attention to details such as mathematics signs and operands, algorithmic steps, and specific clues in math word problems.
Language
Vocabulary and phonological awareness at age 4 predicts math achievement at age 7.
Problems in reading correlate with math disabilities.

20. Interventions for mathematics disabilities
Core classroom instruction:
Elemental instruction:
Classroom instruction should be explicit, academically focused and foster high levels of engagement and frequent opportunities for student response and feedback
Connecting Math Concepts (Engelmann, Carnine, Engelmann, & Kelly, 1991). Highly structured lessons involving frequent teacher questions and student answers.
Mathematical PALS (L. S. Fuchs et all, 1997). Classwide peer-tutoring program designed as a supplement othe classroom teacher’s basal program. Goals include differentiate instruction and provide more intensive practice.
Teach verbal self-instruction.

21. Interventions for mathematics disabilities
Supplemental instruction:
Small group tutoring focusing on word problem intervention:
Pirate Math (L. S. Fuchs et al, 2009). Focuses on x just like pirates focuson on finding treasure on a map.
Identifies types of work problems: combine, compare, and change problems
Explicit and systematic, ric in concepts, incorporates systematic practice, cumulative review and systematic reinforcement.
Arithmetic intervention:
Weakness in counting associated with math deficits
Transition from counting to math fluency

22. Interventions for mathematics disabilities
Fractions interventions:
Half of middle and high school students in the US are not proficient with the ideas and procedures taught about fractions in the elementary grades.
Competence in fractions is considered foundational for learning algebra, for success with more advanced mathematics and for competing successfully in the American workforce.
Most successful interventions utilize explicit instruction. Some focus on procedures while others focus on concepts.
Fraction Face-Off (L. S. Fuchs, Schumacher, Malone, & Fuchs, 2015).
The problem of transfer. Transfer may occur from math facts tutoring to procedural calculations, indicating a connection between these two types of calculation competence. Math facts tutoring does ot transfer to work problem performance. Math curriculum comprises multiple math components within and across grades.

23. Fundamental principles for teaching mathematics LD
1. Teach different components explicitly (e.g., fact retrieval, procedures, problem solving) for whole numbers and rational numbers.
2. Teach arithmetic (math facts) in the context of number knowledge principles to support understanding and strategic counting.
3. Provide speeded strategic practice, with immediate corrective feedback on errors, to build long-term associates in memory and encourage automatic retrieval of answers.
4. For procedural computations, explicitly teach the most efficient algorithms as rules, while providing the conceptual basis for why those procedures work. Begin with worked examples, gradually transferring responsibility to the learners, while providing practice with corrective feedback and cumulative review across problem types.

24. Fundamental principles for teaching mathematics LD
5. For word problem solving, teach problem types (e.g., combine problems, compare problems), introducing one problem type at a time, but systematically providing cumulative review across all taught problems types and practice in sorting problems into problem types. For each problem type, begin by providing the conceptual basis for the problem type; then explicitly teach the most efficient solution strategy, with worked examples and gradual transfer of work to the learner.
6. Explicitly teach for transfer by explaining the ways in which problems may look novel but still represent the taught problems types.
7. Promote self-regulation and independence to promote generalization.

25. Caveats to intervention
Interventions have have not been designed or systematically tested.
Not all students respond.
Effects of tutoring diminish as students continue in school.
More research is needed.

26. References
Engelmann S., Carnine, D. W., Engelmann, O., & Kelly, B. (1991). Connecting math concepts. Chicago: Science Research Associates.
Fletcher, J. M., Lyon, G. R., Fuchs, L. S., & Barnes, M. A. (2018). Learning disabilities: from identification to intervention. Guilford Press.
Fuchs, L.S., Fuchs, D., Hamlett, C.L., Phillips, N.B., Karns, K., & Dutka, S. (1997). Enhancing students helping behavior during peer-mediated instruction with conceptual mathematical explanations. Elementary School Journal, 97,
Fuchs, L. S., Schumaker, R. F., Malone, A. M., & Fuchs, D. (2015). Fraction challenge. Manual available at
Henderson, A(. 2012). Dyslexia, dyscalculia and mathematics: a practical guide. Routledge, Kegan & Paul.

27. Dyscalculia Websites and videos