Transcript Mathematics Instruction for Children with Fetal Alcohol

Mathematics Instruction
for Children with
Fetal Alcohol
Spectrum Disorder:
A Handbook for Educators
Carmen Rasmussen, PhD
Katy Wyper, BSc
Department of Pediatrics
University of Alberta
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: (780) 735-7907, [email protected]
Chapter Overview
1.
Mathematics Deficits in Children with FASD (p. 1)
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2.
General Strategies for Teaching Children with FASD (p. 8)
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3.
Preparing to Teach Students with FASD
Specific Classroom Interventions
Helpful Educational Strategies
Behavioral Interventions
Stages of Math Development (p. 18)
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4.
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
Characteristics of Students with Math Difficulties (p. 24)
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5.
Overview
Math for Students with Disabilities
Assessment of Math Difficulties
Language Ability and Math Difficulties
Strategies for Teaching Children with Math Difficulties (p. 36)
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6.
Children with PAE
Adolescents with PAE
Preschool Children with PAE
Conclusions
Goals of Instruction
Student Centered Approach
General Considerations
Helpful Tips
Teaching Problem-Solving
Strategies for Teaching Children with FASD Math (p. 49)
1) Mathematics Deficits in
Children with FASD
Children with Prenatal Alcohol Exposure
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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.
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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.
1
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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.
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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.
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Of all the achievement subtests, math was the lowest score among
both the alcohol exposed groups, but not the control group.
2
Adolescents with Prenatal Alcohol Exposure
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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.
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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.
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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.
3
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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.
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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.
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.
4
Preschool Children with Prenatal Alcohol Exposure
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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.
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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.
5
Conclusions
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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
6
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
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
Kable et al., in press
7
2) General Strategies for
Teaching Children with FASD
Preparing to Teach Students with FASD
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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 include:1
1)
Collect information to understand the student’s strengths and
weaknesses.
•
look at the student’s history, previous report cards,
psychological reports, IPPs, as well as family and medical
background
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talk with the child about their interested, concerns, and
supports
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talk with the parents about the child’s strengths and
weaknesses
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observe the child in the classroom to evaluate needs and
strategies for support
2)
Make a plan to determine what the child’s 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
3)
Evaluate the plan to determined what is and is not working.
1)
If the child is till having difficulties may need to make a referral for
assistance (classroom aide).
8
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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 child’s current skill level and his or
her specific academic needs.
Functional classroom assessments may also be useful to
understand the child’s 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 child’s functioning.
Important characteristics to observe:
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skills
attention
independence
social interactions
language
strengths and interests
behavior
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Specific Classroom Interventions
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Kalberg and Buckley2 also suggest some specific classroom
interventions for children with FASD:
1) Structure and Systematic Teaching
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structure environment and teaching and teach functional routines so
child knows what is coming next and what is expected
2) Visual Structure
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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
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keep environment simple with few distractions
have obviously defined work areas
4) Task Structure
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structure tasks so that a child understands the task expectations, what
steps to do, and what to do when finished
5) Cognitive Control Therapy
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help each child to understand their own learning style and learning
difficulties
6) Involve the Family
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listen to the families desires and wants
10
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The acronym ‘SCORES’ has been used to depict characteristics of
a good classroom environment for students with an FASD:3
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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)
Finally Danna Ormstup4 suggested the following acronym to make
your classroom ‘ROCK’ for a child with FASD:
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R – Routine
O – Organized
C – Consistency
K – Knowledge base (know each child’s strengths and weaknesses)
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Helpful Educational Strategies
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Wescott5 provides general strategies for educating children with
FAS and FAE:
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slow down and simply information
structure, use consistent daily routines
don’t 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:
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have a calm and quiet environment, using calm colors
minimize distractions and objects hanging on walls
use structure and routines, with simple and consistent rules
headphones may assist with quiet activities
help child prepare for transitions, give breaks throughout the day
use teaching methods that stimulate all senses
use concrete examples and picture calenderers
give children choices
always have the same activity in the same place
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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 include:1
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organizing the classroom with few distractions
color code student’s 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
don’t 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:
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teach within the child’s cognitive and social aptitudes
teach towards each child’s unique learning style and strengths, and
what they are able to do
use small steps and few words
use hands on materials and concrete examples, encourage participation
in activities
use observation learning techniques, imitation, and physical aids
practice, and give feedback
13
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Evensen9 suggests 12 important elements for success when
teaching students with FASD:
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2.
3.
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7.
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10.
11.
12.
encourage success for children with FASD
interact with the child’s 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
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McLaughlin, Williams, and Howard10 review classroom behavioral
interventions for children prenatally exposed to alcohol and
drugs which include:
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Contingency management
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Token reinforcement programs: reinforce positive behaviors
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Contingency contracting: write out contingencies with
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Daily report cards: are also helpful when combined with a home
with tokens, either concrete (chips) or symbolic (points). Use
tokens with social praise.
behaviors and consequences on a contract for both the child
and educator to sign.
reinforcement system.
Behavioral Self-Management
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Self monitoring: in which the child does self-assessment and
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Self-instructional training: self-instruction to improve on-task
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Self-managed drill and practice: Cover, Copy, and Compare
self-recording. This can improve attention, task completion, and
reduce problematic behaviors
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.
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
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Direct Instruction
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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 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.
16
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
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)
7
3) Stages of Math
Development
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According to the UK National Numeracy standards, by the end
of the first year of formal math education, children should be
able to:1
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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
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The Stages of Early Arithmetical Learning (SEAL) model is the
most basic aspect of the LFIN. It describes stages in the
development of children’s arithmetical ability. According to
SEAL, development is characterized by the three parts:
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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
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Early Arithmetical Strategies:
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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 countdown 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.
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Base-Ten Arithmetical Strategies:
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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.
19
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Part B
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FNWS, BNWS, and Numeral Identification:
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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 child’s ability to count a sequence
of number words forwards and backwards, not only by
ones but by other units as well.
Johansson2 suggests that children’s 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:
1.
2.
3.
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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 child’s 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
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Other Aspects of Early Arithmetical Learning:
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These aspects are not as directly addressed by the LFIN but are
nevertheless related to components of parts A and B.
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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
child’s 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: Children’s 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.
21
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:
1.
2.
3.
4.
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repertoire
distribution
efficiency
selection
Compared with typically developing children, children who have
mathematical disabilities in the first and second grades:
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strategy
strategy
strategy
strategy
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.
Intervention should be directed towards the procedural skills that
are lacking in children with MD. Further, it is possible that there
are underlying cognitive factors, such as working memory, that
contribute to the development of mathematical strategies. Finally,
the children examined in this study had mathematical and reading
disabilities, and the results therefore cannot be generalized to
populations without the combination of these disabilities.
22
References
1.
2.
3.
Wright, Martlund, & Stafford, 2000
Johansson 2005
Torbeyns et al.
23
4) Characteristics of Students
with Math Difficulties
Overview
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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
24


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
25
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).
26

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:



improved calculation fluency and accuracy
improved counting strategies (more sophisticated and efficient)
beginnings of a number sense
27

Some suggestions for interventions include:3











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
acknowledging areas of weakness and allowing for more practice and
time spent working in these areas
the transition from concrete to abstract math concepts is imperative in
the development of calculation fluency
28
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 higherlevel 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:






identifying what the problem is asking
picking out the relevant details
choosing the appropriate procedure to solve the problem
estimating a solution
calculating the solution
checking the solution
29



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 student’s 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 student’s 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
30

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 1+3, 2+2, 3+1, and 4+0)
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
31

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.
32
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.
33




It is possible that the discrepancy between informationprocessing 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, children’s 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, children’s attitudes and feelings towards math, and
interactions with other students.
34
References
1.
2.
3.
4.
5.
Chiappe
Hanrahan
Gersten et al
Chap 12
Fazio 1999
35
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 problemsolving.
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.
36



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.
Distinguishing between students with learning difficulties and
learning disabilities is also important in order to provide the
most appropriate instruction. Children with learning
disabilities have no trouble generalizing strategies to other
related areas of learning; however children with learning
difficulties have more problems making such generalization.2
37
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 instructor’s role is to guide the student
to a more solid understanding of math by combining new
knowledge with the student’s 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.
38


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 schooltaught strategies tend to be fairly rigid, it is important to
emphasize flexibility.
39
General Considerations
Some important points to remember when providing instruction:4

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:









Simplicity: There are many different ways to “adjust,”
“modify,” or “adapt” instruction. However, it is best to keep
things simple.




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
use only one or two differentiation strategies in the classroom at
once
only when necessary
be sure to return to your regular teaching style and curriculum as
soon as possible
It is important to ensure that modifications are only
temporary, as lower-achieving students will not benefit from
constantly receiving lessons that are less challenging than
higher-achieving students. Decreasing the demands placed on
lower-ability students may further widen the gap between
lower-achieving and higher-achieving students.
40

CARPET PATCH: A mnemonic device which summarizes
methods that teachers may use to implement differentiation.












Other helpful strategies:







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
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
provide opportunity for extra practice for those students who need
it
extension work for more able students
Students with disabilities: all of the strategies described
above may be appropriate for students with disabilities. In
addition, useful information may be accessed in the student’s
individual education plan (IEP).
Explicit and direct instruction is often the most useful for
students with learning disabilities or difficulties. Practical,
hands-on activities, group work and verbal discussion about
math are also important in the facilitation of math learning.
Group work is most useful when each student has an
opportunity to contribute.
41
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-overhand 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.
42

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.
43
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:






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:






reading the words
understanding specific words within the problem
comprehending the problem in general
linking an appropriate strategy to the problem
interpret the target problem
identify strategies needed to solve the problem
change the problem into an appropriate algorithm
perform computations
evaluate the solution
We also self-monitor and self-correct throughout the entire
problem-solving process. Students with learning difficulties
should be taught these skills through direct instruction and
explanation early on in order to become independent problemsolvers later on. Several techniques teachers may use are:



modeling and demonstration of the appropriate strategies used to
solving routine and non-routing problems
talk through the steps that should be taken, and questions that
should be asked during the entire process
evaluate the steps and procedures used to solve the problem once
it has been completed
44

The use of mnemonics may be useful to teach students a
particular strategy. For example, RAVE CCC:





Once these steps have been taken, CCC outlines what should
follow:





R – Read carefully
A – Attend to key information that gives clues about necessary
procedures
V – Visualize the problem
E – Estimate a potential solution
C – Choose numbers
C – Calculate a solution
C – Check this solution (cross reference with your estimate)
Ideally, as students become more comfortable with problemsolving procedures and strategies, teachers may move from
direct instruction to less-involved guided practice and
eventually the student may become and independent problemsolver.
The use of calculations does not impede students’ progression
from basic number sense, to computational skill, to problemsolving proficiency. In fact, the use of a calculator may allow
teachers to focus more on teaching higher-level problemsolving strategies, and it has even been suggested that
students who use calculators develop more positive feelings
about math.
45

Other techniques teachers may use to facilitate problemsolving 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 should consider each individual student’s
existing knowledge when planning problem-solving lessons.
Intervention strategies that are aimed at a child’s particular
difficulty would likely be most effective. Components of
arithmetic that have been identified by teachers and
researchers as particularly important are related to counting,
the use of written symbols, place value and derived fact
strategies, word problems, relations between concrete, verbal
and numerical forms of problems, estimation, and
remembering number facts.
In terms counting, young children most often encounter
problems with regard to the order-irrelevance issue, and
repeated addition and subtraction by one. Problems in these
areas are improved by practicing counting and cardinality
questions starting with very small numbers and working up.
46




Children’s understanding of written symbols can be solidified
by having the child practice reading and writing simple
arithmetic equations. Place value can be more clearly taught
by presenting children with different forms of addition
including written numbers, number lines and blocks, physical
objects (hands, fingers, blocks), currency (pennies and
dimes), and any kind of mathematical apparatus. To clarify
children’s understanding of word problems a useful technique
is to present addition and subtraction word problems, and
discuss their characteristics with the child.
The relation between concrete, verbal, and numerical forms of
arithmetic problems appears to cause particular difficulty
among children. To resolve this difficulty, it has been found
useful to present the similarities among different forms and
demonstrate why each form has the same answer.
Derived fact strategies can be taught by presenting two similar
arithmetic problems to children, teaching an effective strategy
for solving one of the problems, and then explaining how and
why the same strategy may be used for the second problem.
Lessons on estimation are often successful when children are
asked to judge estimates made by make-believe characters.
That is, children are shown a group of arithmetic problems as
well as proposed answers (given by pretend characters), and
asked first to evaluate the answers and then provide a
justification for their evaluation. Finally, memory for number
facts can be improved by repeatedly presenting children with
simple arithmetic facts (e.g. 2 + 2 = 4) over multiple sessions
and playing games to strengthen memory for these facts.
Both teachers and students who have tested these
intervention techniques deemed them useful and fun, and a
valuable way to spend one-on-one time with each other.
Further, a particularly meaningful outcome of these
intervention strategies is that children often gained selfesteem and confidence in their mathematical abilities.
47
References
1.
2.
3.
4.
Chap 12
Milo 2005
Lucangeli
Chap 13
48
6) Strategies for Teaching
Children with FASD Math

Kvigne, Struck, Engelhart and West1 give specific
recommendations for teaching children with FAS/FAE
mathematics:





a child with FAS/FAE may be able to count but may not understand
what the numbers mean, so instructors must teach what each
number means
allow the child to see, hear, and feel each number (using tactile
stimulation)
teaching functional math skills (time and money) is important
allow children to use their fingers or calculators to aid with addition
and subtraction
When teaching children with FASD, it is important to develop
the student’s understanding number concepts, math facts,
computations, mental problem solving and word problems.2

It is also suggested to focus on math vocabulary, directionality,
functional uses of math, and time and money concepts2 .

Other teaching strategies specifically for math include2 :










using number lines and concrete examples and materials
practice math facts
allowing more time on tests and assignments
don’t have too many problems on one page
keep similar types of problems together on the same page
allow child to use calculators, graph paper, and highlighters to
signify start and end points
focus on practical and functional math
promote organization
made use different types of technologies (computers, tapes)
use a variety of teaching strategies
49

Some of the deficits children with an FASD may show in math
which include difficulties with3:









understanding what numbers mean and represent
remembering basic math facts and symbols
understanding the direction of a particularity math problem
too many problems on a page
knowing when to use a concept and what strategies or step are
required to solve the problem
reading word problems
money concepts
sequencing numbers
Consequently, it is suggested that when teaching children with
FASD mathematics it important to3:












determine what level the child is at and don’t presume that they
have successfully learned all the concepts taught in prior years
focus on basic math skills
teach in a slow pace, use repetition and practice
use clear and concise examples
Because children with an FASD may have difficulty understanding
symbols, numbers, and concepts and also how to use them in
different contexts, it is important to use consistent language when
teaching (e.g., ‘plus’ can also be said as ‘sum’, ‘and’, ‘add’ etc.).
teach specific operations in the same way
write down steps in multi-step problems and use checklists
ensure that they truly understand a concept
allow extra practice time
computer math programs may help
use diagrams and visuals, and start at the easiest level
special emphasis understanding time
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References
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Kvigne, Struck, Engelhart and West
Manitoba Manual
C and O Manual
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