Mathematical Disabilities

1
Mathematical Disabilities

• ??????

2
Some Facts in the United State

• 6 of school-age children have some type of
  learning order in mathematics (Badian, 1983
  Kosc, 1974)
• 70 of them are boys (Badian, 1983)

3
Cognitive Deficit in MD (mathematically disabled)
children

• Retrieving facts from long-term memory
• executing arithmetical procedures
• Many MD children appear to outgrow the procedural
  deficit, but the memory-retrieval problems tend
  to be persistent (Geary et al., 1991)

4
Cognitive Components

• When solving arithmetic problems,
• they tend to use immature problem-solving
  strategies
• have rather long solution times
• frequently commit procedural and memory-retrieval
  errors
• make poor strategy choices

5
Relationship between performance on achievement
and ability tests and contributing cognitive
skills
Conceptual Knowledge
Procedural Skills
Working Memory
Test Performance
Working Memory
Knowledge Base Fact Retrieval
Counting Speed
6
Differences between Normal and MD Children
Achievement test at the end of Kindergarten year
Cognitive Addition Task
At the end of 1st grade
1st grade normal
1 st grade MD-improved
1st grade MD
1st grade MD-no change
7
Follow-up study (at the end of 1st grade)

• Same types of problem-solving strategies
  (retrieval, verbal counting or counting fingers)
• MD-no change group
• high frequency of procedural errors
• frequent use of immature counting-all procedure
• a lot of variability in the speed of executing
  the counting and retrieval strategies

• Normal children
• increased reliance on fact retrieval
• made fewer counting and fact-retrieval errors
• faster
• MD-no change
• no change in the mix of strategies
• no change in the number of facts remembered
• high rate of errors
• considerable improvement in the use of counting
  procedures

8
Conclusion

• Procedural skills of many MD children are
  developmentally delayed
• fact-retrieval problems probably represents a
  more fundamental deficit that many of these
  children might not completely outgrow.

9
Conceptual Knowledge

• MD children
• had an immature understanding of the essential
  and unessential features of counting, e.g.,
  adjacency was essential for correct counting.
• Poor at detecting certain forms of counting
  errors
• procedural skills of MD children may be due to an
  immature understanding of the associated
  concepts, e.g., carrying

10
Working Memory

• MD children perform worse in tasks requiring
  retaining information in working memory
• contribute to
• their poor procedural skills
• remembering arithmetical facts.

11
Speed of Processing

• On average, MD children take longer to solve
  arithmetic problems
• slower at executing basic numerical processes?
• Use a different mix of problem-solving
  strategies?
• Geary (1990) 1 grade children, no diff. In
  counting speed, but MD slower in reading,
  speaking numbers...
• Slower counting speed for MD children, 2nd grade,
  8- 9- old.
• 4th grade, no diff.

12

• Unlikely counting speed is a contributing factor
• older MD children take longer to retrieve facts
  from memory
• mental slowness -- use of more time-consuming
  problem-solving strategies

13
Neuropsychological Component

• It is rare to find case studies in which a focal
  brain injury produces a distinct mathematical
  deficit
• general categories of acquired and developmental
  dyscalculias
• alexia and agraphia for numbers
• spatial acalculia
• anarithmetria

14
Alexia and Agraphia for Numbers

• Difficulties in the reading and writing of
  numbers, intact skills in other areas of
  arithmetic
• associated with lesions of the left hemisphere
• sometimes associated with reading and other
  language-related disorders

15
Spatial Acalculia

• Difficulties in the spatial representation of
  numerical information
• associated with damage to the posterior regions
  of the right hemisphere
• problems include number omissions, number
  rotations, misreading arithmetical operation
  signs, difficulties with place value and
  decimals, misalignment of numbers in multicolumn
  arithmetic problems

16

• A disturbance of visuospatial functions will
  affect the ability to spatially represent
  numerical information and skills at interpreting
  the meaning of these representations (e.g., place
  value) while leaving reading and other
  language-related skills relatively intact.

17
Anarithmetria

• Difficulty in the retrieval of basic arithmetic
  facts from long-term memory
• associated with damage to the posterior regions
  of the left hemisphere
• number reading and writing and the spatical
  representation of numerical information are
  typically intact.
• Understanding of arithmetical concepts is intact

18
Genetic Component

• MD and RD coexist in many children
• RD children with auditory memory deficits often
  have problems with arithmetic memory functions
• arithmetic and mathematics performance may rise
  to average or above-average levels when the words
  used in the problems are minimized or learned by
  sight.
• RD children with phonological processing and
  auditory memory deficits often have a specific
  deficit in arithmetic-fact retrieval
• retrieval of arithmetic facts and verbal
  information in semantic memory are similar

19

• many forms of RD appear to be heritable
• It is quite possible that MD is also heritable

20
Mathematical Disabilites, Subtype 1 Semantic
Memory

• Cognitive/Performance Features
• Low frequency of arithmetic-fact retrieval.
• When facts are retrieved, there is a high error
  rate.
• Solution times for correct retrieval are
  unsystematic
• Developmental
• Appears to follow a developmental difference
  model. Performance is qualitatively distinct from
  that of normal children and shows minimal
  grade-to-grade improvement.

21

• Neuropsychological Features
• Appears to be associated with left-hemispheric
  dysfunction, in particular posterior regions of
  the left hemisphere.
• Possible subcortical involvement, such as the
  thalamus
• Genetic Features
• Unclear, but the relationship with certain forms
  of RD suggests that this deficit might be
  heritable
• Relationship to RD
• Often covaries with RD, especially if the RD is
  associated with phonetic deficits.

22
Mathematical Disabilities, Subtype 2 Procedural

• Cognitive/Performance Features
• Relatively frequent use of developmentally
  immature procedures
• Frequent errors in the execution of procedures
• Potential developmental delay in the
  understanding of the concepts underlying
  procedural use.
• Developmental Features
• Appears to follow a developmental delay model.
  Performance is similar to that of younger,
  academically normal children and improves from
  grade to grade.

23

• Neuropsychological Features
• Unclear, although some data suggest an
  association with left-hemispheric dysfunction
• Genetic Features
• Unclear
• Relationship to RD
• Unclear

24
Mathematical Disabilities, Subtype 3 Visuospatial

• Cognitive/Performance Features
• Difficulties in spatially representing numerical
  information, such as misaligning numbers in
  multicolumn arithmetic problems or rotating
  numbers.
• Misinterpretation of spatially represented
  numerical information, such as place-value errors
• Developmental Features
• Unclear

25

• Neuropsychological Features
• Appears to associated with right-hemispheric
  dysfunction, in particular, posterior regions of
  the right hemisphere.
• Genetic Features
• Unclear
• Relationship to RD
• Does not appear to associated with RD, at least
  not with the forms of RD that are associated with
  phonetic deficits.