Mathematics Instruction for Children with Fetal Alcohol Spectrum Disorder: A Handbook for Educators

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Mathematics Instruction for Children with Fetal
Alcohol Spectrum DisorderA Handbook for
Educators

• Carmen Rasmussen, PhD
• Katy Wyper, BSc
• Department of Pediatrics
• University of Alberta

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• The development of the manual was funded by
  the Alberta Centre for Child, Family, and
  Community Research

Correspondence concerning this manual should be
addressed to Carmen Rasmussen Department of
Pediatrics, University of Alberta 137 GlenEast,
Glenrose Rehabilitation Hospital 10230-111Ave,
Edmonton, Alberta, T5G 0B7 Phone (780)
735-7999, ext 15631 Fax [phone],
carmen_at_ualberta.ca
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Chapter Overview

• Mathematics Deficits in Children with FASD (p. 1)
• Children with PAE
• Adolescents with PAE
• Preschool Children with PAE
• Conclusions
• General Strategies for Teaching Children with
  FASD (p. 8)
• Preparing to Teach Students with FASD
• Specific Classroom Interventions
• Helpful Educational Strategies
• Behavioral Interventions
• Stages of Math Development (p. 18)
• Learning Framework in Number
• Part A Early Arithmetic Strategies Base-Ten
  Arithmetic Strategies
• Part B Forward Number Word Sequences, Backward
  Number Word Sequences, Numerical Identification
• Part C Other Aspects of Early Arithmetic
• Strategy Competence

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1) Mathematics Deficits in Children with FASD

• Children with Prenatal Alcohol Exposure
• The most direct evidence for the effect of
  prenatal alcohol exposure on mathematics
  difficulties among offspring comes from the
  landmark longitudinal study by Streissguth, Barr,
  Sampson, and Bookstein1.
• Over 500 parent-child dyads participants, with
  about 250 of the mothers classified as heavier
  drinkers and about 250 as infrequent drinkers or
  as abstaining from alcohol (based on maternal
  report of alcohol use during mid-pregnancy).
• From preschool to adolescence, these children
  were tested on a variety of outcome variables
  including IQ, academic achievement,
  neurobehavioral ratings, cognitive and memory
  measures, and teacher ratings.
• Of all these outcome variables, performance on
  arithmetic were the most highly correlated with
  prenatal alcohol exposure at age 42, 7 years3,
  114, and 145. Thus, the more alcohol these
  children were exposed to, the poorer they did on
  tests arithmetic, and this relation with alcohol
  exposure was the strongest of all of the
  variables measured.
• Furthermore, 91 of the children who performed
  poorly on arithmetic at age 7 were still low at
  age 14, highlighting the stability and robustness
  of this finding. For older children maternal
  binge drinking appeared to be most related to
  lower arithmetic performance.
• The authors Streissguth5 highlighted the
  recurrent finding that arithmetic is especially
  difficult for individuals who were prenatally
  exposed to alcohol.

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• In a study of 512 mother-child dyads,
  Goldschmidt6 examined the relation between
  maternal report of alcohol use during pregnancy
  and academic achievement of offspring at 6 years
  of age.
• The authors found that drinking during the second
  trimester was related to difficulties in reading,
  spelling, and arithmetic. Furthermore, after
  controlling for IQ, prenatal alcohol exposure was
  still significantly related to arithmetic but
  only marginally related to reading and spelling.
  This indicates that these substantial deficits in
  arithmetic can not be solely attributed to a low
  IQ.
• Others have found that 7-year-olds with prenatal
  alcohol exposures have a slower processing speed
  and a specific deficit in processing numbers.7
• Furthermore, arithmetic is one of the only
  measures that differentiates children with
  FAS/FAE from those with ADHD, in that only those
  with FAS/FAE show deficits in arithmetic.8
• In another study, Coles9 examined the cognitive
  and academic abilities of children aged 5 to 9
  years from three groups a control group not
  exposed to alcohol a group whose mothers stopped
  drinking during the second trimester and group
  whose mothers drank throughout the pregnancy.
• Of all the achievement subtests, math was the
  lowest score among both the alcohol exposed
  groups, but not the control group.

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• Adolescents with Prenatal Alcohol Exposure
• Arithmetic deficits have also been documented in
  adolescents with FASD.
• Streissguth et al. 10 found that adolescents and
  adults with FAS/FAE performed the poorest on
  arithmetic scoring at the second grade level for
  arithmetic, third grade for spelling, and fourth
  grade for reading.
• Furthermore, adults with FAS, both with average
  and below average IQ, have been found to score
  lowest on the arithmetic tests (as compared to
  other academic areas) and only arithmetic scores
  were lower than predicted based on IQ. 11
• Kopera-Frye12 specifically examined number
  processing among 29 adolescents and adults (aged
  12 to 44) with FAS/FAE and control participants
  matched on age, gender, and education level.
• Participants were tested on number reading,
  number writing, and number comparison tests as
  well as exact and approximate calculation of
  addition, subtraction, and multiplication. They
  also completed a proximity judgment test in which
  they were to circle one of two given numbers that
  was about the same quantity as the target number
  (e.g., 15 17 or 27).
• Participants also completed a cognitive
  estimation test in which they were presented with
  questions for which they had to provide a
  reasonable estimate, such as what is the length
  of a dollar bill? or how heavy is the heaviest
  dog on earth? Before testing judges determined
  what would be the acceptable range for guesses.
• The group with FASD made significantly more
  errors than the controls on cognitive estimation,
  proximity judgement, exact calculation of
  addition, subtraction and multiplication, and
  approximate subtraction.

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• Furthermore, the highest number of participants
  was impaired on cognitive estimation, followed by
  approximate subtraction. Although the FASD group
  tended to answer with the correct units of
  measurement (feet, pounds) on the cognitive
  estimation test, their range of answers was far
  broader than those of the controls. For example,
  one participant answered 5 feet for the length of
  a dollar bill.
• Hence, despite having intact number reading,
  writing, and comparison skills, the participants
  displayed deficits in many other areas of number
  processing, particularly calculation and
  cognitive estimation.
• Using a similar math battery with 13-year-olds,
  Jacobson et al.13 found that prenatal alcohol
  exposure was related to deficits in exact
  addition, subtraction, and multiplication,
  approximate subtraction and addition, and
  proximity judgment and number comparison.
• Two main factors emerged calculation (exact and
  approximate) and magnitude representation (number
  comparison and proximity judgment). Thus it
  appears that the math deficits evident in FASD
  may be in two different areas, one relating more
  to calculating and the other involved in
  estimation and magnitude representation.
• Finally, Howell14 compared academic achievement
  of adolescents with prenatal alcohol exposure,
  controls children, and special education
  students. The special education group had poorer
  overall achievement, as well as in reading and
  writing, but still those with prenatal alcohol
  exposure were significantly impaired in
  mathematics.
• Mathematics deficits have even been reported in
  Swedish adolescents with prenatal alcohol
  exposure.

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• Preschool Children with Prenatal Alcohol Exposure
• Little research has been conducted on math
  abilities in preschool children prenatally
  exposed to alcohol.
• Kable and Coles15 looked at the relation between
  prenatal alcohol exposure and math and reading in
  4-year-old children from a high-risk (high
  alcohol exposure) and low-risk (low alcohol
  exposure) groups and found that the high-risk
  group performed significantly lower than the low
  risk-group on math but not reading.
• In a recent study, Rasmussen Bisanz,16 examined
  the relation between mathematics and working
  memory in young children (aged 4 to 6 years of
  age) diagnosed with an FASD.
• Children with FASD displayed significant
  difficulties on the two mathematics subtests
  (applied problems and quantitative concepts)
  which measure problem solving, and knowledge of
  math terms, concepts, symbols, number patterns,
  and sequences.
• Age was negatively correlated with performance on
  the quantitative concepts subtest, indicating
  that older children performed worse, relative to
  the norm, than younger children on the
  quantitative concepts subtest. Thus quantitative
  concepts appear to be particularly difficult with
  age among children with FASD.
• Moreover, children with FASD performed well below
  the norm on measures of working memory, which
  were correlated with math performance indicating
  that the math difficulties in children with FASD
  may result from underlying deficits in working
  memory.

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• Conclusions
• There is considerable evidence indicating that
  children and adolescents with FASD and prenatal
  alcohol exposure have specific deficits in
  mathematics and particularly arithmetic.
• These findings have been consistent across a
  multitude of both longitudinal studies and group
  comparison studies, even after controlling for
  many confounding variables and IQ. Thus, these
  math deficits are not simply due to a lower IQ
  among those with FASD, but rather prenatal
  alcohol exposures appears to have a specific
  negative affect on mathematics abilities.
• More research is now needed to determine why
  children with FASD have such deficits in
  mathematics and what area of mathematics are most
  difficult for these children, which is important
  to modify instruction and tailor intervention to
  improve mathematics.
• There is very little intervention research among
  children with FASD, and even less intervention
  research on mathematics and FASD.
• However, recently Kable,17 developed and
  evaluated a math intervention program for
  children aged 3 to 10 years with FAS or partial
  FAS. The math intervention program included
  intensive, interactive, and individual math
  tutoring with each child. It also focused on
  cognitive functions such as working memory and
  visual-spatial skills that are involved in
  mathematics.
• Children were assessed before and after the 6
  week intervention, and after the intervention
  children in the math intervention group showed
  more improvements in math performance than
  children not in the math intervention.
• This is the first study to demonstrate
  improvements in math among children with an FASD
  and future research is needed to examine the
  long-term efficacy of such and intervention, the
  most appropriate duration of such a program, as
  well whether such positive benefits can be
  observed in group classroom settings.18

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• References
• Streissguth, Barr, Sampson, and Bookstein (1994)
• Streissguth, 1989
• Streissguth, 1990
• Olson, 1992
• Streissguth, 1994
• Goldschmidt (1996)
• Burden (2005)
• (Coles 1997
• Coles 1991
• Streissguth et al 1991
• Kerns, 1997
• Kopera-Frye, (1996
• Jacobson et al. (2003)
• Howell, (2006
• Kable and Coles (2003, April
• Rasmussen Bisanz, 2007
• Kable, (in press

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2) General Strategies for Teaching Children with
FASD

• Preparing to Teach Students with FASD
• Children with FAS/FAE difficulties in social
  emotional, physical, and cognitive functioning
  (particularly learning, attention sequencing,
  memory, case and effect reasoning, and
  generalizations).1
• Some suggestions for preparing to teach children
  with FAS/FAE include1
• Collect information to understand the students
  strengths and weaknesses.
• look at the students history, previous report
  cards, psychological reports, IPPs, as well as
  family and medical background
• talk with the child about their interested,
  concerns, and supports
• talk with the parents about the childs strengths
  and weaknesses
• observe the child in the classroom to evaluate
  needs and strategies for support
• Make a plan to determine what the childs needs
  to be successful.
• look at resources, manuals, handbooks
• consult with other teachers and special
  education teachers, professionals, counsellors,
  and psychologists.
• develop activities to focus on the most important
  needs of the child
• Evaluate the plan to determined what is and is
  not working.

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• Kalberg and Buckley2 suggest that when developing
  an Individualized Program Plan (IPP) for a child
  with FASD it is important to also evaluate each
  childs current skill level and his or her
  specific academic needs.
• Functional classroom assessments may also be
  useful to understand the childs real life
  abilities. The authors suggest observing each
  child in different natural settings on a few
  different occasions to understand conditions that
  both disrupt and enhance each childs
  functioning.
• Important characteristics to observe
• skills
• attention
• independence
• social interactions
• language
• strengths and interests
• behavior

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• Specific Classroom Interventions
• Kalberg and Buckley2 also suggest some specific
  classroom interventions for children with FASD
• 1) Structure and Systematic Teaching
• structure environment and teaching and teach
  functional routines so child knows what is coming
  next and what is expected
• 2) Visual Structure
• individualized visual schedules, routines, visual
  organizations
• visual instructions and visual cues
• color coding, labelling areas of classroom and
  tasks
• highlight important information on a task
• ensures the environment and tasks are clear and
  predicable and helps with child with sequencing
  events, transitions, anticipating what will come
  next
• 3) Environmental Structure
• keep environment simple with few distractions
• have obviously defined work areas
• 4) Task Structure

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• The acronym SCORES has been used to depict
  characteristics of a good classroom environment
  for students with an FASD3
• S Supervision, Structure, Simplicity
• C Communication, Consistency
• O Organizations
• R Rules (simple and concrete)
• E Expectations (realistic and attainable)
• S Self-esteem (acceptance and encouragement)

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• Helpful Educational Strategies
• Wescott5 provides general strategies for
  educating children with FAS and FAE
• slow down and simply information
• structure, use consistent daily routines
• dont overload them with stimuli
• keep transitions constant
• reduce words and verbal cues
• focus on real life skills
• set up simulated stores, banks, etc in the
  classroom
• use metaphors with concrete choices
• use visual cues (pictures, cartoons) to depict
  daily activities
• promote good coordination and communication
  between parent and teacher
• Kvigne et al.6 suggest many helpful education
  techniques for children with FAS/FAE including
• have a calm and quiet environment, using calm
  colors
• minimize distractions and objects hanging on
  walls

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• Burgess and Streissguth7 describe children with
  FAS/FAE as impulsive, having difficulty with
  transitions, poor judgment, not understanding
  consequences, and poor communication. They
  suggest guiding educational principles such as
  early intervention, focus on communication skills
  and making choices, and teaching social skills.
  To manage problematic behaviors the authors
  suggest teaching communication skills and making
  choices, planning ahead, and creating a balance
  between structure and independence.
• Other general teaching strategies for children
  with FASD include1
• organizing the classroom with few distractions
• color code students material in one binder
• use pictures
• have quite work areas
• use moderate lighting and heating and warm colors
• have structure and consistent rules
• prepare the child a head of time for changes and
  transitions
• have simple rules with consistent consequences
• avoid too many choices
• dont give too much homework
• find a medium between not expecting too little
  and expecting too much
• Sobsy8 provide a list of instructions tips for
  children prenatally exposed to drugs and alcohol
  which include
• teach within the childs cognitive and social
  aptitudes

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• Evensen9 suggests 12 important elements for
  success when teaching students with FASD
• encourage success for children with FASD
• interact with the childs family and respect
  their emotions
• try a different approach when things are not
  working
• structure
• observe behaviour
• interpret behaviour
• ensure the environment is meeting the sensory
  needs of the child
• use concrete language
• know the memory difficulties faced by children
  with FASD
• recognize their social and academics difficulties
• appreciate the life transitions for individuals
  with FASD
• reinforce and praise success

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• Behavioral Interventions
• McLaughlin, Williams, and Howard10 review
  classroom behavioral interventions for children
  prenatally exposed to alcohol and drugs which
  include
• Contingency management
• Token reinforcement programs reinforce positive
  behaviors with tokens, either concrete (chips) or
  symbolic (points). Use tokens with social praise.
• Contingency contracting write out contingencies
  with behaviors and consequences on a contract for
  both the child and educator to sign.
• Daily report cards are also helpful when
  combined with a home reinforcement system.
• Behavioral Self-Management
• Self monitoring in which the child does
  self-assessment and self-recording. This can
  improve attention, task completion, and reduce
  problematic behaviors
• Self-instructional training self-instruction to
  improve on-task behavior. For example, a teacher
  demonstrates a problem and solution, and then the
  student performs the task while saying the steps
  out loud, then whispering, then silently.
• Self-managed drill and practice Cover, Copy, and
  Compare technique,11 Student looks at problem,
  covers problem, answers problem, uncovers
  problem, and compares their response to the
  original. This technique is private and allows
  the child to work at his or her own pace. It is
  effective with children with behavior problems
  and disabilities, and has been successfully used
  in math.

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• Peer Tutoring
• Classwide peer tutoring helpful for students
  with poor achievement or who have disadvantages
  pr disabilities.
• Reciprocal peer tutoring students service as
  both tutor and tutee
• Cross-age peer tutoring having older student
  assist younger students
• Direct Instruction
• Direct instruction involves numerous
  student-teacher interactions, well-planned and
  sequenced lessons, and modern learning
  techniques.12 The aim is to teach more in less
  time by teaching in small groups in a fast pace,
  using a few examples that can be applied to a
  number of different situations, and giving
  instant and positive corrections.10
• McLaughlin et al.10 also suggest that because
  medications can be effective with student with
  ADHD they may be useful when combined with
  behavioral interventions for children with
  prenatal exposure to alcohol and drugs, but more
  research is needed on these effects.

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• References
• Manitoba Manual
• Kahlberg and Buckley (2007)
• C and O Manual
• Danna Ormstup (March, 2007)
• Wescott (1991)
• Kvigne, Struck, Engelhart and West
• Burgess and Streissguth (1992)
• Sobsy
• Deb Evensen, (2007, March)
• McLaughlin, Williams, and Howard (1998)
• McLaughlin and Skinner (1996)
• Engelmann and Carnine (1998)

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3) Stages of Math Development

• According to the UK National Numeracy standards,
  by the end of the first year of formal math
  education, children should be able to1
• accurately count 20 objects
• count forward and backward by ones from any small
  number and count by tens from zero and back to
  zero
• read, write and understand the order and
  vocabulary of numbers 0 through 20
• understand the operations used in addition and
  subtraction, and the associated vocabulary (e.g.
  take away)
• remember all number pairs that have a total of
  ten
• say the number that is one or ten larger or
  smaller than any other number from 0 to 30
• Learning Framework in Number (LFIN)1
• The Stages of Early Arithmetical Learning (SEAL)
  model is the most basic aspect of the LFIN. It
  describes stages in the development of childrens
  arithmetical ability. According to SEAL,
  development is characterized by the three parts
• Part A. Early Arithmetic Strategies Base-Ten
  Arithmetical Strategies
• Part B. Forward Number Word Sequences (FNWS)
  Number Word After Backward Number Word Sequences
  (BNWS) Number Word Before Numeral
  Identification
• Part C. Other Aspects of Early Arithmetical
  Learning

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• Part A
• Early Arithmetical Strategies
• Emergent Counting children are unable to count
  visible objects due to either not knowing words
  for numbers or not being able to coordinate the
  words with the objects.
• Perceptual Counting children are able to count
  perceived (ie. heard, seen, or felt) objects, but
  not objects in a screened collection
• Figurative Counting children can count objects
  in a screened collection but this counting is
  still rudimentary (e.g. when asked to add two
  collections and told how many object are in each,
  children count objects one by one instead of
  counting on from the largest screen.)
• Initial Number Sequence children are now able
  to count-on and to solve addition problems with
  one number missing (e.g. 4 _ 7). Children
  can also use some count-down strategies (e.g.
  15 4 as 14, 13, 12, 11).
• Intermediate Number Sequence children are able
  to use count-down strategies more efficiently.
• Facile Number Sequence children can now use a
  range of strategies not limited to counting by
  ones (e.g. recognizing that there is a 10 in all
  teen numbers).
• Once children have advanced to Stage 6, they
  progress through 3 levels involving the use of
  base-ten strategies.
• Base-Ten Arithmetical Strategies
• Level 1 Initial Concept of Ten Children can
  count to and from 10 by ones but do not recognize
  ten as a unit.
• Level 2 Intermediate Concept of Ten Children
  now recognize 10 as a unit, but cannot perform
  any operations on it without the components being
  represented in groups of ones (e.g. two open
  hands) they cannot perform operations on tens in
  the written form.
• Level 3 Facile Concept of Ten Children are
  now able to solve addition and subtraction
  problems without material representations.

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• Part B
• FNWS, BNWS, and Numeral Identification
• Number words are the spoken and heard names of
  numbers (Wright, et al). The LFIN draws an
  important distinction between a child being able
  to actually count and being able to recite a list
  numbers in the correct order. Knowledge of
  forward and backward number order sequences is a
  childs ability to count a sequence of number
  words forwards and backwards, not only by ones
  but by other units as well.
• Johansson2 suggests that childrens knowledge of
  number words is related to other numerical
  abilities. For example, children may recognize a
  structure in number word sequences and use this
  structure to solve arithmetic problems. There
  are three levels a child goes through to when
  learning how to arithmetics
• the child uses physical objects to represent
  addends (e.g. David has 3 apples and Simon has 2
  apples. How many apples are there?)
• the child uses non-physical representations to
  solve problems (e.g. verbal unit items)
• the child uses known facts or procedures to solve
  problems
• Numerals are the written and read form of
  numbers. Numeral identification is a childs
  ability to produce the name of a given numeral.
  Identification is different from recognition in
  that to recognize, a child must simply pick out a
  named numeral among a random set as opposed to
  producing the name him or herself.

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• Part C
• Other Aspects of Early Arithmetical Learning
• These aspects are not as directly addressed by
  the LFIN but are nevertheless related to
  components of parts A and B.
• Combining and Partitioning Children may learn to
  recognize combinations and partitions of numbers
  (e.g. one and four is five seven is three and
  four). These sets of numbers become automotized
  so that children have knowledge of them without
  having to count one by one.
• Spatial Patterns and Subitizing This aspect
  involves a childs ability to recognize spatial
  patterns such as dominos patterns, playing card
  patterns, or dot cards. To subitize is a
  technical psychological term which means to
  capture the number of dots in a stimulus without
  actually counting them.
• Temporal Sequences These are stimuli, such as
  sounds or movements, that occur sequentially
  time.
• Finger Patterns Childrens use of fingers
  strategies increases in complexity as they
  advance through the stages of SEAL. Eventually
  it is expected that children will no longer rely
  on their fingers, but these strategies play a
  very important role in early stages.
• Base-Five (Quinary-Based) Strategies Base-five
  strategies are useful in situations that involve
  sets of five items.

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• Strategy Competence
• In a study of children with math and reading
  difficulties, Torbeyns et al.3 concluded that
  strategy competence develops along the following
  four dimensions
• strategy repertoire
• strategy distribution
• strategy efficiency
• strategy selection
• Compared with typically developing children,
  children who have mathematical disabilities in
  the first and second grades
• have the same strategy repertoire (retrieval,
  counting)
• use retrieval less
• use more immature forms of counting
• are slower at selecting strategies
• implement strategies less accurately
• make less adaptive strategy choices
• Most of these differences between MD and typical
  children seem to decrease with age, however
  strategy frequency characteristics remain.
  Children with MD show less strategy development
  than typical children (e.g. they continue to rely
  on counting strategies, while typical children
  use retrieval at an increasing frequency) and
  these differences may exist as a result of a
  developmental delay instead of a developmental
  deficit. That is, the mathematical abilities of
  children with MD develop more slowly than those
  of typical children, but they eventually develop
  nonetheless.

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• References
• Wright, Martlund, Stafford, 2000
• Johansson 2005
• Torbeyns et al.

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4) Characteristics of Students with Math
Difficulties

• Overview
• According to Chiappe,1 math difficulties (MD)
  appear to be the consequence of a specific
  deficit rather than a general learning problem.
  If MD were a result of some general deficit,
  those children with problems in math would also
  experience problems in other areas, but this is
  not the case. Two factors that may be responsible
  for MD some children encounter are problems with
  number representation and the inability to
  process numerical stimuli. Longitudinal research
  provides support for the latter.
• Studies have documented the existence of number
  representation and processing as early as infancy
  and early childhood. 1 Interruptions in the
  normal development of these processes may be the
  cause of math deficits found in older children.
  An improper representation of number can cause
  difficulties in counting, number sense, and
  discriminating quantities. For example, some
  children are able to count from one to five, but
  do now know whether 4 is greater than 2 or 2 is
  greater than 4.1

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• Children with learning problems have difficulties
  describing what they are thinking when they added
  numbers.2 However, they use strategies similar
  to those used by typical children when adding
  numbers (count-all, and count-on, with or without
  the use of physical objects). This suggests
  that, similar to typically developing children,
  children with learning problems do in fact
  acknowledge relationships between numbers instead
  of simply depending on rote memorization when
  performing addition problems.
• One issue to be aware of is that sometimes
  students may provide a correct answer to a math
  problem by using the wrong strategy. It is
  important to keep this in mind, because it could
  easily go unnoticed in a classroom situation.2
  It has been documented that sometimes children
  try to hide their hands while counting on their
  fingers. Due to the fact that students with
  learning problems may never pass the point of
  depending on physical objects to count, it is
  important to encourage the use of these objects
  when performing math problems. 2

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• Math for Students with Disabilities3
• Students that have difficulties with math in
  elementary school seem to have more problems
  retrieving number facts in higher grades. This
  difficulty perpetuates into upper level math such
  as algebra.
• Counting strategies
• another difference that shows up between students
  with and without math difficulties is the
  complexity of their counting strategies
• young students with math difficulties may use the
  same strategies as students without difficulties,
  but they tend to make more mistakes
• the strategies that students use to count are a
  good predictor of how receptive they will be to
  traditional teaching techniques
• Reading difficulties seem to exacerbate the
  problems that students encounter in mathematics.
• One of the primary deficits in students with math
  difficulties is poor calculation fluency
  (recalling number facts quickly and relying on
  simple strategies).

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• Number sense
• Defined as
• fluency in estimating and judging magnitude
• ability to recognize unreasonable results
• flexibility when mentally computing
• ability to move among different representations
  and to use the most appropriate representation
• Two indicators of number sense in young children
  are counting ability and quantity discrimination.
  Quantity discrimination may be associated with
  informal math learning that occurs outside of the
  school setting, whereas counting may be more
  dependent on formal education
• Number sense may be used to predict future
  performance in other areas of math, the first
  four of which are influenced by instruction
• quantity discrimination/magnitude comparison
• missing number in a sequence
• number identification
• rapid naming
• working memory
• Early intervention should focus on

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• Some suggestions for interventions include3
• encouraging student to depend on their retrieval
  skills as opposed to counting
• technologies that allow individualized practice
• instruction focusing on strategy development and
  use
• automatization of number facts and teaching
  shortcuts
• improves both number sense and fluency
• small group work that promotes familiarity and
  comfort with numbers
• developing math vocabulary
• structured peer work
• using visuals and multiple representations
• teaching strategies that could be used as a
  hook for problem-solving

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• Assessment of Math Difficulties4
• Problems that students with special needs often
  encounter while learning math include
• inadequate or unsuitable instruction
• curriculum that is too fast-paced
• lack of structure which promotes discovery
  learning
• teachers use of language that does not math
  students level of understanding
• early use of abstract symbols
• trouble reading math word problems (students with
  reading difficulties)
• problems with basic math relationships which
  propagate into higher-level math
• insufficient revision of early learned math
  concepts
• In order to avoid simply watering-down the math
  curriculum for students with learning
  difficulties, is may be useful to incorporate
  math in other areas of learning such as social
  studies, sciences, reading, and writing.
• The first step towards fostering a more solid
  understanding of math in students with
  difficulties is to determine what they already
  know, identify any holes that may exist, and
  formulate a plan to fill these holes. This may
  be done by constructing mathematical skills
  inventories which reflect the curriculum to be
  taught. Teachers may keep track of the types of
  mistakes students are making, and use these
  patterns to identify weaknesses.
• Informal interviews between teacher and student
  may also be a useful technique to identify skills
  and weaknesses. Several areas that are important
  in problem-solving ability are

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• Asking questions like why did the student have
  trouble with this area?, would the use of
  concrete objects or other aids help the student
  solve this problem? and is the student able to
  explain to me what to do? may help determine the
  extent of difficulty, and where exactly the
  misunderstanding occurs in the problem-solving
  process.
• To build on a students existing knowledge, it
  must first be determine how much the student
  knows. Assessment can be broken down into three
  Levels
• Level 1 The student has trouble with basic
  number. First, examine the students vocabulary
  of number relationships and conservation of
  number. Assessment must then be done by
  examining each of the following items in order
• sort by a single attribute
• sort by two attributes
• create equal sets using one-to-one matching
• count objects to ten, then twenty
• recognize numerals to ten, then twenty
• correctly order number symbols to ten, then
  twenty
• write down spoken numbers to ten, then twenty
• understand ordinality (first, seventh, fourth,
  etc)
• add numbers below ten with counters and in
  writing
• subtract numbers below ten with counters and in
  writing
• count-on in addition
• solve simple oral addition and subtraction
  problems (numbers below ten)
• familiarity with coins and paper currency

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• Level 2 Performance is slightly higher than in
  Level 1. Assess the following
• mental addition below twenty
• mental problem-solving without using fingers or
  tally-marking
• mental subtraction is there a discrepancy
  between addition and subtraction performance?
• vertical and horizontal written addition
• understanding of addition commutativity (i.e. the
  order of addends does not matter) does the
  student always count-on from the largest number?
• understanding of additive composition (every
  possible way of producing a number e.g. 4 is
  13, 22, 31, and 40)
• understanding of the complementary order of
  addition and subtraction problems. For example,
  7 3 4 3 4 7 and 5 3 2 5 2 3.
• translate an operation observed in concrete
  objects to a written equation
• transfer a written equation into a concrete
  equation
• translate a real-life scenario into a written
  problem and solve it
• recognize and write numbers up to fifty
• tell digital and analogue time
• list the days of the week
• list the months of the year

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• Level 3 The student is able to perform most of
  the item in Level 1 and 2
• read and write numbers to 100, then 1000
• read and write money additions
• mentally compute halves or doubles
• perform mental addition of money determine
  amounts of change using count-on
• memorize and recite multiplication tables
• add hundreds, tens, units and thousands,
  hundreds, tens, units with and without carrying
• know the place values with thousands, hundreds,
  tens, units
• subtraction algorithm with and without exchanging
  columns
• correctly perform the multiplication algorithm
• correctly perform the division algorithm
• understand fractions
• correctly read and solve basic word problems
• Translating abstract concepts into tangible,
  concrete problems is helpful for children with
  learning disabilities. It is important however,
  to ensure that students do not learn to rely on
  these physical objects, and that they gradually
  transition from concrete to abstract
  understanding.

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• Language Ability and Math Difficulties5
• Children with specific language impairment (SLI)
  appear to have difficulties in counting and
  knowledge of basic number facts, however they are
  quite successful on written calculations with
  small numbers. One area that may cause trouble
  for students with SLI is the increased amount and
  complexity of mathematical vocabulary these
  children are exposed to in higher elementary
  school (grades 4 and 5). This presents a problem
  because children with SLI have a tough time
  retrieving information that has been rote
  memorized. Another area in which children with
  SLI show difficulty is information-processing and
  this difficulty can produce challenges with the
  recall of declarative knowledge, and procedural
  knowledge. The mathematics required of upper
  elementary school students demands a combination
  of conceptual, procedural and declarative
  knowledge all of which present problems for
  children with SLI.
• Students with SLI are poorer at recalling number
  facts as well as using correct procedures for
  problem solving. They tend to rely more on
  simple strategies like counting and less on
  advanced strategies like retrieval.
• Children with SLI perform better on written
  calculation tasks when they are un-timed,
  suggesting that these children are indeed capable
  of performing well, just at a slower pace than
  typically developing children. Written
  calculation task performance was much worse when
  children were timed. Tasks that are performed
  under a time constraint tend to load on working
  memory, which ties in to why children with SLI
  would show difficulties on such problems.

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• It is possible that the discrepancy between
  information-processing abilities in typically
  developing children and children with SLI may be
  explained in part by the improved automaticity in
  typically developing children. If true, children
  with SLI who are given the opportunity to
  practice may show improvements in their own
  automaticity, thus freeing up cognitive resources
  that could be used for other processes.
  Moreover, childrens performance on timed tasks
  should improve if they are taught strategies to
  automatize because they can spend less time tasks
  that were once controlled and consciously
  attended to. Two ways in which automatization
  might be encouraged are computer-based
  interventions and paper-and-pencil drill and
  practice games.
• Another factor that may play role in the
  difficulty that children with SLI encounter when
  it comes to math problems is that many of these
  children are living in poverty and often receive
  poorer education that children from a more
  affluent family.
• Children with SLI experienced may problems with
  the procedural aspect of calculations. The
  author suggests two ways to rectify this problem
  (1) by encouraging students to think through
  the steps involved in answering a particular
  question, and (2) instructing children to ask
  themselves questions such as what operation must
  I use for this problem? Teaching students to
  confirm their answers to math problems (e.g. 87
  24 63, 63 24 87) may help them understand
  mathematical concepts and relationships.
• Finally, childrens attitudes and feelings
  towards math, and interactions with other
  students.

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• References
• Chiappe
• Hanrahan
• Gersten et al
• Chap 12
• Fazio 1999

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5) Strategies for Teaching Children with Math
Difficulties

• Goals of Instruction1
• There are five goals of mathematics education to
  learn the value of mathematics, to build
  confidence in mathematic ability, to learn how to
  solve mathematical problems, to learn how to
  communicate mathematically, and reason
  mathematically.
• Students proficient in math possess the following
  skills
• Conceptual understanding understanding of
  concepts, relations, and operations.
• Procedural fluency perform procedures with
  skill, speed, and accuracy.
• Strategic competence develop appropriate plans
  for problem-solving.
• Adaptive reasoning the ability to think about
  problems flexibly and from different
  perspectives.
• Productive disposition enjoying and appreciating
  math, and being motivated to improve mathematical
  ability.

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• It is important to distinguish between and
  identify math difficulties and disabilities,
  because the identification and intervention may
  prevent children with math weaknesses from
  developing a full disability.
• New amendments to American legislation have
  recently been made in a project called IDEA.
  These modifications are geared towards helping
  children with learning disabilities as well as
  their families and teachers. Three areas that
  are affected by the amendments are criteria for
  determination of eligibility, whether the child
  will respond to research-based intervention, and
  the percentage of federal funds that may be
  allotted to early intervention services.

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• Student-Centered Approach
• It was once believed that math should be taught
  in the form of rule-based instruction, whereas
  now, research supports a more student-focused
  form of instruction. That is, teachers should
  consider students existing mathematical
  knowledge and provide an environment in which
  realistic problems combine with and strengthen
  this existing knowledge. This process is called
  Realistic Mathematics Education (RME).2
• According to Milo et al.2 one responsibility of
  the teacher is to facilitate knowledge
  construction based on the students existing
  knowledge. One kind of instruction is guiding
  instruction
• Guiding instruction the instructors role is to
  guide the student to a more solid understanding
  of math by combining new knowledge with the
  students own contributions (guiding instruction)
  as opposed to simply directing the students about
  mathematical concepts (directing instruction).
  In guiding instruction, students are encouraged
  to reflect upon new strategies that they learn,
  which teaches them to choose more appropriate
  strategies in the futures.
• However, students with special needs may not
  benefit from this type of instruction.
  Generally, students with learning problems have
  difficulties structuring the strategies that they
  learn. Consequently, a more directive
  instructional approach may be more appropriate
• Directing instruction the teacher provides the
  student with explicit rules and structure may
  reduce the ambiguity that sometimes exists in
  guiding instruction.

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• In the directing instruction, one specific
  strategy may be taught in isolation, as opposed
  to guiding instruction, where students are
  encouraged to compare and choose (based on their
  own existing knowledge) among multiple
  strategies, and then to explain their choices.
  Typically-developing children may benefit most
  from guiding instruction, while children with
  special needs benefit more from directing
  instruction. The use of supporting models (e.g.
  number lines, number position schemes) also
  contribute to the special needs students
  understanding of appropriate and effective
  strategy use.
• Children may tend to rely more on strategies
  formally learned in school and less on strategies
  they may have learned before entering school.3
  Children also show overconfidence in these
  strategies, regardless of their effectiveness.
  Because school-taught strategies tend to be
  fairly rigid, it is important to emphasize
  flexibility.

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• General Considerations
• Some important points to remember when providing
  instruction4
• Differentiation recognize differences among
  individual students and modify instruction
  according to these differences. This method may
  be used with students who have disabilities or
  learning problems, and also those who are the
  most gifted. Examples
• personalized learning objectives for each student
• adapting curricula to suit the students
  cognitive level
• different paths of learning for different
  learning styles
• spend more or less time on lessons depending on
  students rates of learning
• modifying instructional resources (manuals,
  texts)
• allow the students to produce work through a
  variety of media
• be flexible with grouping students
• adjusting the amount of help or guidance giving
  to each student
• Simplicity There are many different ways to
  adjust, modify, or adapt instruction.
  However, it is best to keep things simple.
• use only one or two differentiation strategies in
  the classroom at once
• only when necessary

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• CARPET PATCH A mnemonic device which summarizes
  methods that teachers may use to implement
  differentiation.
• C curriculum content
• A activities
• R resource materials
• P products from lessons (what students are
  asked to produce)
• E environment
• T teaching strategies
• P pace
• A amount of assistance
• T testing and grading
• C classroom groupings
• H homework assignments
• Other helpful strategies
• re-teach some concepts using different language
  and examples
• use different techniques to maintain interest of
  less motivated students
• modify the amount and detail of feedback given to
  students

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• Helpful Tips1
• Counting. Sometimes children will learn to
  memorize counting rhymes, but not connect these
  rhymes with the actual counting of physical
  objects. Guidance (hand-over-hand or direct,
  explicit teaching) may help students to make this
  connection, which is so fundamental in early math
  learning.
• Numerals. Familiarity and recognition of numerals
  may be fostered by repetition presentation in the
  form of flash cards or other games.
  Over-learning gives lower-ability students the
  chance to establish a solid base on which they
  can build higher math skills.
• Written numbers. Children with learning
  difficulties may have problems if introduced to
  written number symbols too early. A good
  alternative is to use dot schemes, tally marks,
  or other number representations before using
  number symbols.
• Number Facts. Another area of weakness for some
  students with learning difficulties is the
  automatic retrieval of number facts (e.g. 4 2
  6) as well as knowledge about mathematical
  procedures (what to do when you see ).
  Ensuring that students learn facts and
  computational procedures through increased
  regular practice and number games will allow them
  to solve math problems more quickly and easily.
  Calculators can also be used to aid students with
  computational difficulties, but some teachers may
  not wish to substitute traditional written math
  with an electronic device.
• Number Games. Instead of having children
  complete traditional exercises and worksheets,
  turn math learning into a game. Using small
  candies or toys can make lessons interesting and
  fun, but it is important to make sure that these
  lessons remains educational, not just
  entertaining.

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• Where Next?1
• Once students form a solid knowledge base of
  numbers and counting, lessons may be advanced to
  actual computation in the horizontal and vertical
  forms. When a student is learning these
  procedures, it is important that they receive
  consistent help from teachers, aides, and
  parents. The same language, cues, and steps
  should be used so that the student does not
  become confused. However, it is also important
  to teach students a variety of techniques to
  solve these problems, particularly ones which
  will help the student learn more about number
  structure and composition.
• It has been shown that adults rely more on
  addition and subtraction in every day life than
  multiplication and division, so if a teacher must
  prioritize math curriculum, it may be useful to
  focus most on addition and subtraction, followed
  by multiplication, and finally division.
• Students with perceptual problems may require
  slight modifications in teaching material in
  order to perform on paper-and-pencil problems.
  Some examples that may be useful are thick
  vertical lines, squared paper, and small arrows
  or dots that the students may follow on the page.

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• Teaching Problem-Solving1
• The next step in math learning, problem-solving,
  could be a particularly difficult task for
  students with disabilities because they may have
  trouble in the following areas
• reading the words
• understanding specific words within the problem
• comprehending the problem in general
• linking an appropriate strategy to the problem
• Consequently, students may feel overwhelmed or
  hopeless when attempting such problems and it is
  important to teach them how to feel confident and
  comfortable working through these problems.
• People generally problem-solve in the following
  order
• interpret the target problem
• identify strategies needed to solve the problem
• change the problem into an appropriate algorithm
• perform computations
• evaluate the solution

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• The use of mnemonics may be useful to teach
  students a particular strategy. For example,
  RAVE CCC
• R Read carefully
• A Attend to key information that gives clues
  about necessary procedures
• V Visualize the problem
• E Estimate a potential solution
• Once these steps have been taken, CCC outlines
  what should follow
• C Choose numbers
• C Calculate a solution
• C Check this solution (cross reference with
  your estimate)
• Ideally, as students become more comfortable with
  problem-solving procedures and strategies,
  teachers may move from direct instruction to
  less-involved guided practice and eventually the
  student may become and independent
  problem-solver.
• The use of calculations does not impede students
  progression from basic number sense, to
  computational skill, to problem-solving
  proficiency. In fact, the use of a calculator
  may allow teachers to focus more on teaching
  higher-level problem-solving strategies, and it
  has even been suggested that students who use
  calculators develop more positive feelings about
  math.

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• Other techniques teachers may use to facilitate
  problem-solving competence in students with
  learning difficulties include
• teaching difficult vocabulary before-hand
• using cues to show students where to begin and
  where to go from there (e.g. arrows)
• connecting problems with students own life
• allowing students to create problems and have
  other people solve them
• encouraging the use self-monitoring and
  self-correction
• As always, teachers shoul