Difficulties and Disabilities in Learning Mathematics Summit on Math and Science Education October 4

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Difficulties and Disabilities in Learning
Mathematics Summit on Math and Science
EducationOctober 4, 2007Association of
American PublishersSchool Division

• Daniel B. Berch, Ph.D.
• Child Development and Behavior Branch
• National Institute of Child Health
• and Human Development

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Why is learning even basic math skills so
important?

• As information becomes ever more quantitative
  and as society relies increasingly on computers
  and the data they produce,

an innumerate citizen today
is as vulnerable as the illiterate peasant of
Gutenbergs time.
Lynn Arthur Steen (1997) Why Numbers Count
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School-Entry Math Skills Predict Achievement
in Later Grades

• Duncan et al. (in press) conducted a coordinated
  analysis of six longitudinal data sets relating
  changes in early skills to later teacher ratings
  and test scores of reading and math achievement,
  holding constant important background
  characteristics. Findings
• School-entry math, reading, and attention skills
  were associated with later achievement.
• Early mathematical skills had the most striking
  predictive power and was a much more powerful
  predictor of later reading achievement than early
  reading was of later math achievement.
• Early math skills predicted later reading
  achievement as well as early reading achievement
  did!

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Basic Math Skills Are Important in the
Workplace

• Analyzing data from two longitudinal studies,
  Murnane and Levy (1995) found that basic math
  skills (i.e., fractions, decimals, line graphs)
  of high school seniors, taught in U.S. schools by
  the eighth grade, are predictive of their wages
  at 24 years of age.

• In their 1996 book, Teaching the New Basic
  Skills, Murnane and Levy note that the basic
  math skills required in the average American
  workplace are actually those which ought to be
  learned by the ninth grade yet many American
  students graduate from high school without
  mastering them.

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Many of the seventh graders I teach have a poor
sense of numbers. They dont understand that
adding two numbers results in a larger number,
that multiplication is repeated addition, that 5
x 6 is larger than 5 x 4 or that one-quarter is
smaller than one-half. This lack of basic math
facts detracts from their ability to focus on the
more abstract operations required in math at a
higher level.
Susan B. Sheridan


Washington Post,

December 27, 2004
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Assessing Disposition

• Math is like a jungle. The ideas are all jumbled
  up.

• Learning math is like exploring an unknown
  country.
• You make lots of choices, where to go, what to
  do.

• Doing a math problem is like finding your way
  through a
• maze. There are lots of possible pathways to
  go down.

Fennell (2004)

• Math class is tough! Mattels Barbie (1992)

-- which is usually followed by
Lets go shopping!
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A popular actress and sometime mathematician has
leveraged shopping and related activities to
demonstrate the value of math to middle-school
girls
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Shopping can be challenging if one lacks
basic math skills.
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Effective financial as well as workplace skills
can be seriously hampered when one suffers from a
learning disability in mathematics, as described
in the following testimonials
Work is incredibly difficult as I've lost jobs
because I can't count properly. It somehow
implies untrustworthiness if someone doesn't want
to handle money or avoids using figures. (JB)
The only time I break out into a cold sweat is
when a potential client asks me for a quote on
an editing project (I do freelance English
language proofreading/editing). I always have to
tell them I'll get back to them in a couple of
hours. That's how long it takes me to multiply
the number of pages by the fee that I plan to
charge! I always seem to get a wrong total each
time, even using a calculator. Often I email a
colleague and ask him/her what he/she would
charge, and then take it from there.

(Dyscalculia Forum
participant)
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Mathematical Learning Difficulties Vs.
Disabilities

• There is no consensus definition of mathematical
  learning disabilities (MD).

• Nevertheless, use of the term disability
  implies a biologically-based disorder
  characterized by specific cognitive deficits.

• Estimates of the prevalence of MD range from 5
  to 10.

• Diagnosis of MD in research is usually based on
  a cutoff score at or below the 20th or 25th
  percentile on a standardized mathematics
  achievement test.

• Issue of stability Many children who score
  below average on math achievement tests during
  one academic year score average or better in
  subsequent years. Such children should be
  classified as experiencing mathematical learning
  difficulties due to factors other than a learning
  disability.

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(Some) Sources of Math Difficulties (other
than an underlying disability)

• Novelty of newly introduced math principle,
  concept
• or procedure

• Absence from school

• Midyear relocation to a new school

• Societal or cultural factors

• Language of instruction

• Characteristics of the curriculum

• Characteristics of the teacher

Mazzocco (2007)
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Domains in Which Children With MD Exhibit
Deficiencies

• Counting knowledge

• Working memory

• Long-term memory

• Immediate recognition of numerosities

• Estimation skills

• Fractions and decimals

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Counting Deficits in Children With MD

• Many children with MD, independent of their
  reading
• achievement levels or IQ, have a poor
  conceptual
• understanding of some aspects of counting.
• These children understand most of the inherent
• counting rules, such as stable order and
  cardinality,
• but they consistently err on tasks that assess
  order
• irrelevance or adjacency.
• The poor counting knowledge of these children
• appears to contribute to their delayed
  competencies in
• the use of counting to solve arithmetic
  problems and
• may result in poor skill at detecting and,
  thus,
• correcting counting errors (e.g., double
  counting).

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Counting Knowledge
Children with MD failed to detect a double
counting error on 1 of 3 trials
Percent Correct ID
MD Math Disabled LA Low Achieving TA
Typically Achieving
Geary (2006)
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Working Memory

• This is most simply characterized as the
  workspace
• of the mind.
• This is a limited capacity system that
  simultaneously
• stores and processes information
• This system controls information storage and
• processing by means of a hypothesized central
• executive through attention and
  inhibition.
• The information is represented and stored in
  either a
• language (phonological) system or a
  visuospatial
• system.

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Working Memory
Standard Score
Geary (2006)
M 100, SD 15
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Retrieval Deficits in Children With MD

• The most persistent (and often defining) deficit
  
• typically observed in MD is the ability to
  store and
• retrieve number combinations/facts from
  long-term
• memory.
• This fact retrieval deficit manifests itself in
  the form
• of slow and unpredictable response times.
• Lack of automaticity indicating poor retrieval
  also
• means that limited working memory resources
• are directed toward the use of often slow and
  
• inefficient counting strategies and away from
  the
• more complex aspects of mathematical
  processing.

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Individuals with MD are also slower and less
efficient at recognizing the numerosities of
visual displaysusually dots (Butterworth
Reigosa, 2007).
. . . .


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Naming and Ordering Fractions and Decimals
0.2 0.05 7/10 0.6 10/10
.002 20/100 7/100 0.50 60/100
Smallest
Biggest
Circle tied quantities (e.g.,
)
0.2
20/100

• Longitudinal study with middle schoolers (6th
  and 8th grade)

• Children with low math achievement (LA) but no
  learning disability can accurately name (i.e.,
  orally read) decimals, possess at least some
  ability to retrieve facts about fractions and
  decimals, but fail to correctly rank order
  intermixed decimals and fractions.

• Children with MD are unable to accurately name
  decimals, fail to identify equivalent
  representations of ratios (e.g., 0.5 ½),
  misidentify incorrect equivalents, such as 0.05
  0.50, and fail to correctly rank order decimals
  or fractionsalone or intermixed.
  
  

Mazzocco Devlin (in press)
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Think It Through
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Estimation Study
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6th graders
2nd graders
0
100
Siegler Opfer (2003)
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Mental Number Line
Linear Representation
1 2 3 4 5 6
7 8 9

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71
6th graders
2nd graders
0
1000
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Mental Number Line
Logarithmic Representation
1 2 3 4
5 6 7 89
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Number Line Estimation in MD Children
Percent Use
Strategy
Geary (in press)
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The mental number line and number sense
The mental number line underlies our intuitive
understanding of numbers, or what has been
referred to as number sense.
Gersten Chard (1999) have suggested that number
sense is as important to the learning of
mathematics as phonemic awareness is to learning
to read.
How is number sense defined?
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Features of Number Sense

• Number sense is difficult to define but easy to
  recognize.

• Students with good number sense can move
  seamlessly between the real world of quantities
  and the mathematical world of numbers and
  numerical expressions.

• They can represent the same number in multiple
  ways depending on the context and purpose of the
  representation.

• They have a good sense of numerical magnitude
  and can recognize gross numerical errors, that
  is, errors that are off by an order of magnitude.
  

• They can think or talk in a sensible way about
  the
• general properties of a numerical problem or
  expressionwithout doing any precise computation.

Case (1998)
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Predicting First-Grade Math Achievement From
Number Sense Growth

• Tracked number sense development of 277 children
  from the beginning of kindergarten through the
  middle of first grade (number sense battery
  included counting, number knowledge, nonverbal
  calculation, simple story problems, and basic
  arithmetic number combinations) and then
  assessed them on general math achievement at the
  end of first grade (Woodcock-Johnson III Math).
• Number sense performance in kindergarten, as
  well as number sense growth, accounted for 66
  percent of the variance in first-grade math
  achievement.
• Background characteristics of income status,
  gender, age, and reading ability did not add
  explanatory variance over and above growth in
  number sense.
• Children who started kindergarten with low
  number sense but made moderate gains by the
  middle of the kindergarten year had higher
  first-grade math achievement than children who
  started out with similarly low number sense but
  had flat growth (the majority of the latter were
  from low-income families).
• These results suggest that screening early
  number sense growth may be useful for identifying
  children who will face later math difficulties or
  disabilities.

Jordan et al. (2007)
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The NICHD/DoED
Mathematical Cognition
and Specific Learning Disabilities
Research
Consortium

• Ten projects being carried out at universities
  and research institutions across the country and
  in the UK.

• Includes studies of the neurobiological and
  genetic substrates of MD, longitudinal analyses
  of cognitive deficits, studies of MD subtypes,
  math precursor skills in children at risk for
  developing MD, and instructional interventions
  for disabilities in math problem solving.

• Participants children with idiopathic MD
  (i.e., etiology is unknown), co-morbid math and
  reading disabilities, and neurodevelopmental
  disorders (e.g., Turner and Fragile X syndromes).

• This consortium encourages cross-project
  communication and convergence of methods,
  measures, and findings to build the knowledge
  base in this area and inform educational
  practice.

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NICHD/DoED Mathematical Cognition and Specific
Learning Disabilities Research Consortium
Univ of Missouri Geary
Penn State Univ Petrill and Univ of
London, UK Plomin
Univ of Delaware Jordan
UC Davis M.I.N.D. Institute Simon
Johns Hopkins Univ Mazzocco
Stanford Univ Menon
Vanderbilt Univ Fuchs
Florida Atlantic Univ Hecht

• Project Sites

Univ of Houston Fletcher
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E-mail Query to D. Berch
. . . I'm interested in the cognitive and
brain development of kids prior to school age
related to mathematical development.
My background is in mathematics and I am
currently working on my master's thesis related
to metacognition and mathematics learning.
Any help would be much appreciated.
C.R. Instructional Leader, Mathematics XXX
District School Board
"The best teachers teach from the heart, not from
the book."
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The best teachers are those who, by virtue of
having their hearts in the right place, use their
minds to

• make judicious choices of books and other
  curricular materials
• that are founded on a rigorous evidence base,

• tailor instructional approaches to individual
  students by taking
• into account the variations in their cognitive
  abilities, affective
• styles, and other relevant traits and
  proclivities, and

• constantly monitor the impact of these
  approaches through
• formative as well as summative assessments

-- all in an effort to achieve both short- and
long-term instructional objectives that are
challenging yet reasonable, and ultimately
attainable if given the right confluence of
student effort, teacher preparation, parental
cooperation, administrative support, and
community backing.

Berch (2006)
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Extra Slides
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Here are some relevant gray areas


Three parietal circuits activated in number
processing tasks (After Dehaene et al., 2003).
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The Brain and Mathematical Thinking fMRI -
functional Magnetic Resonance Imaging

• - Shows brain activity (indirectly)
• - Takes a series of pictures over time,
  (e.g.,
• one every two seconds)
• - The f in fMRI means functional, that is,
  
• you get a movie of brain function, not a
• still image of brain structure
• - Measures changes in blood oxygenation
• caused by changes in neural activation

http//www.fmrib.ox.ac.uk/image_gallery/av/
Raizada (2006)
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Leo Tolstoy -
A man is like a fraction whose numerator is
what he is and whose denominator is what he
thinks of himself.
The larger the denominator the
smaller the fraction.
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• Title  Mathematical Cognition and Specific
  Learning Disabilities
• Release/Posted Date July 23, 2007
• Purpose To stimulate innovative,
  multidisciplinary research which will contribute
  to our knowledge of the key factors that
  influence the development and expression of
  learning disabilities in mathematics.
• Major objectives
• identify the critical causal factors (e.g.,
  cognitive, neurobiological, genetic)
• develop explicit, measurable, and reliable
  criteria for differentiating mathematical
  disabilities from more transitory math
  difficulties.
• develop and test well-defined, evidence-based
  treatment interventions.
• 
  
  

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Letter From Concerned Parent Excerpts