Title: Mathematics Instruction for Children with Fetal Alcohol Spectrum Disorders: A Handbook for Educators
1Mathematics Instruction for Children with Fetal
Alcohol Spectrum DisordersA Handbook for
Educators
- Carmen Rasmussen, PhD
- Katy Wyper, BSc
- Department of Pediatrics
- University of Alberta, and
- Glenrose Rehabilitation Hospital
2- 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,
carmen_at_ualberta.ca
3Chapter Overview
- Stages of Math Development (p. 1)
- 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. 7) - Overview
- Math for Students with Disabilities
- Assessment of Math Difficulties
- Language Ability and Math Difficulties
- Strategies for Teaching Children with Math
Difficulties (p. 19) - Goals of Instruction
- Student Centered Approach
- General Considerations
- Helpful Tips
- Teaching Problem-Solving
41) 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
1
5- 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 (i.e. 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 (e.g. 7 3 as 8, 9, 10) 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 the last stage of
Early Arithmetical Strategies, 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.
2
6- Part B
- FNWS, BNWS, and Numeral Identification
- Number words are the spoken and heard names of
numbers. 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 forward and
backward, 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 when
learning how to do arithmetic - 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 (recognition) as
opposed to producing the name him or herself
(identification).
3
7- 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 automatized
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 domino 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.
4
8- Strategy Competence
- In a study of children with reading and math
difficulties (MD), 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 patterns 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 will eventually
develop nonetheless.
5
9- References
- Wright, R.J., J. Martland, and A.K. Stafford,
Early Numeracy Assessment for Teaching and
Intervention. 2000, London Paul Chapman
Publishing Ltd. - Johansson, B.S., Number-word sequence skill and
arithmetic performance. Scandinavian Journal of
Psychology, 2005. 46(2) p. 157-167. - Torbeyns, J., L. Verschaffel, and P. Ghesquière,
Strategy development in children with
mathematical disabilities Insights from the
Choice/No-Choice method and the
Chronological-Age/Ability-Level-Match design.
Journal of Learning Disabilities 2004. 37(2) p.
119-131.
6
102) 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 the difficulties some children encounter are
problems with number representation and the
inability to process numerical stimuli. - 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 not know whether 4 is greater than 2 or 2 is
greater than 4.1
7
11- Children with learning problems have difficulties
describing what they are thinking when they add
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 incorrect 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 -
8
12- 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).
9
13- 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 for students with difficulties
should focus on
10
14- Some suggestions for interventions include3
- encouraging student to depend on their retrieval
skills as opposed to counting (e.g. Mad
Minutes, a game in which children must complete
as many simple arithmetic problems as possible in
one minute) - technologies that allow individualized practice
(e.g. computerized math games) - 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 (e.g. teaching
procedures that may be applied across different
problem-solving situations)
11
15- 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 that promotes discovery
learning - teachers use of language that does not match
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, it 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. For example, several skills that
are necessary in problem-solving are
12
16- 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
13
17- 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
04, 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
14
18- Level 3 The student is able to perform most of
the items in Level 1 and 2, and - 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.
15
19- 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 hard 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 and 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, but it is simply at a slower
pace than typically developing children. Written
calculation task performance is much worse when
children are timed. Tasks that are performed
under a time constraint tend to load on working
memory, which may help to explain why children
with SLI would show difficulties on such
problems.
16
20- It is possible that the discrepancy between
information-processing abilities in typically
developing children and children with SLI may be
due in part to 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 on
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 than children from a more
affluent family. - Children with SLI experience many problems with
the procedural aspect of calculations. The
author suggests two ways to improve 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 (e.g. 87 24 63, 63 24
87) may help them develop a better
understanding of mathematical concepts and
relationships. - Finally, childrens attitudes and feelings
towards math, and interactions with other
students, affects their success in math.
17
21- References
- Chiappe, P., How Reading Research Can Inform
Mathematics Difficulties The Search for the Core
Deficit. Journal of Learning Disabilities, 2005.
38(4) p. 313-317. - Hanrahan, J., S. Rapagna, and K. Poth, How
children with learning problems learn addition A
longitudinal study. Canadian Journal of Special
Education, 1993. 9(2) p. 101-109. - Gersten, R., N.C. Jordan, and J.R. Flojo, Early
Identification and Interventions for Students
With Mathematics Difficulties. Journal of
Learning Disabilities, 2005. 38(4) p. 293-304. - Westwood, P., Commonsense methods for children
with special educational needs - 4ed. 2004
RoutledgeFalmer. - Fazio, B.B., Arithmetic calculation, short-term
memory, and language performance in children with
specific language impairment A 5-yr follow-up.
Journal of Speech, Language, and Hearing
Research, 1999. 42(2) p. 420-431.
18
223) 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 to 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.
19
23- 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 disability. - difficulty due to an underlying intellectual
deficit - disability typical intellect, but are often
accompanied by behavioral and/or emotional
problems
20
24- 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 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 future. - 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 which
may reduce the ambiguity that sometimes
accompanies guiding instruction.
21
25- In 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 special
needs students understanding of appropriate and
effective strategy use. - Children may tend to rely more on strategies
formally learned 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 (e.g.
represent one procedure or problem in multiple
ways).
22
26- General Considerations
- Some important points to remember when providing
instruction1 - 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 strategies in the classroom
at once - use these strategies only when necessary
23
27- 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 (e.g. a variety of
visual, hands-on, or verbal approaches) - modify the amount and detail of feedback given to
students
24
28- 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 make this
connection, which is so fundamental in early math
learning. - Numerals. Familiarity and recognition of numerals
may be fostered by repetitive 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 problems 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 remain educational, not just entertaining.
25
29- 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 division1, 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 well 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.
26
30- Teaching Problem-Solving1
- The next step in math learning, problem-solving,
could be a particularly difficult task for
lower-ability students 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 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
27
31- 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 estimation) - 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 will hopefully become an independent
problem-solver. - The use of calculators does not impede students
progression from basic number sense, to
computational skill, to problem-solving
proficiency. In fact, calculators 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.
28
32- Other techniques teachers may use to facilitate
problem-solving competence in students with
learning difficulties include - teaching difficult vocabulary before-hand
29
33- Early Intervention4
- Intervention strategies that are aimed at a
childs specific difficulty are likely to be most
effective. Components of arithmetic identified
by teachers and researchers to be particularly
important are related to - Counting young children most often encounter
problems with order-irrelevance, 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. - The use of written symbols Childrens
understanding of written symbols can be
solidified by having the child practice reading
and writing simple arithmetic equations. - Place value and derived fact strategies 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. 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.
- Word problems To improve childrens
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 this
relation appears difficult for children to grasp.
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.
30
34- Estimation 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. - Remembering number facts 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.
Further, a particularly meaningful outcome of
these intervention strategies is that children
often gained self-esteem and confidence in their
mathematical abilities.
31
35- References
- Westwood, P., Commonsense methods for children
with special educational needs - 4ed. 2004
RoutledgeFalmer. - Milo, B., A. Ruijssenaars, and G. Seegers, Math
instruction for students with special educational
needs Effects of guiding versus directing
instruction. Educational and Child Psychology,
2005. 22(4) p. 68-80. - Lucangeli, D., et al., Effective Strategies for
Mental and Written Arithmetic Calculation from
the Third to the Fifth Grade. Educational
Psychology, 2003. 23(5) p. 507-520. - Dowker, A., Numeracy recovery A pilot scheme for
early intervention with young children with
numeracy difficulties. Support for Learning,
2001. 16(1) p. 6-10.
32
364) 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 was the most highly correlated with
prenatal alcohol exposure at age 42, 73, 114, and
145. Thus, the more alcohol these children were
exposed to, the poorer they did on tests of
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. - Streissguth5 highlighted the recurrent finding
that arithmetic is especially difficult for
individuals who were prenatally exposed to
alcohol.
33
37- 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 exposure 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 a 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.
34
38- 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.
35
39- 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.
36
40- 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) group 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 Bisanz16 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 this subtest.
Thus quantitative concepts appear to become
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.
37
41- 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 exposure 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, Kable17 developed and
evaluated a math intervention program for
children aged 3 to 10 years with FAS or partial
FAS. The 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 program, and after the program 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 an intervention, the
most appropriate duration of such a program, as
well whether such positive benefits can be
observed in group classroom settings.18
38
42- References
- Streissguth, A.P., et al., Prenatal alcohol and
offspring development the first fourteen years.
Drug and Alcohol Dependence, 1994. 36 p. 89-99. - Streissguth, A.P., et al., Neurobehavioral
effects of prenatal alcohol III. PLS analyses of
neuropsychologic tests. Neurotoxicology and
Teratology 1989. 11 p. 493-507. - Streissguth, A.P., H.M. Barr, and P.D. Sampson,
Moderate prenatal alcohol exposure Effects on
child IQ and learning problems at age 71/2 years.
Alcoholism Clinical and Experimental Research
1990. 14 p. 662-6269. - Olson, H.C., et al., Prenatal exposure to alcohol
and school problems in late childhood A
longitudinal prospective study. Development and
Psychopathology, 1992. 4(3) p. 341-359. - Streissguth, A.P., A long-term perspective of
FAS. Alcohol Health Research World, 1994.
18(1) p. 74-81. - Goldschmidt, L., et al., Prenatal alcohol
exposure and academic achievement at age six A
nonlinear fit. Alcoholism Clinical and
Experimental Research, 1996. 20(4) p. 763-770. - Burden, M.J., et al., Effects of prenatal alcohol
exposure on attention and working memory at 7.5
years of age. Alcoholism Clinical and
Experimental Research, 2005. 29 p. 443-52. - Coles, C.D., et al., A comparison of children
affected by prenatal alcohol exposure and
attention deficit, hyperactivity disorder.
Alcoholism Clinical and Experimental Research,
1997. 21(1) p. 150-161. - Coles, C.D., et al., Effects of prenatal alcohol
exposure at school age I. Physical and cognitive
development. Neurotoxicology and Teratology,
1991. 13(4) p. 357-367. - Streissguth, A.P., et al., Fetal Alcohol Syndrome
in adolescents and adults. The Journal of the
American Medical Association, 1991. 265 p.
1961-1967. - Kerns, K.A., et al., Cognitive deficits in
nonretarded adults with Fetal Alcohol Syndrome.
Journal of Learning Disabilities, 1997. 30(6) p.
685-93. - Kopera-Frye, K., S. Dehaene, and A.P.
Streissguth, Impairments of number processing
induced by prenatal alcohol exposure.
Neuropsychologia, 1996. 34(12) p. 1187-96. - Jacobson, S.W., Didge, N., Dehane, S., Chiodo, L.
M., Sokol, R. J., Jacobson, J. L, Evidence for
a specific effect of prenatal alcohol exposure on
number sense. Alcoholism Clinical and
Experimental Research, 2003. 27(121A).
39
43- References (continued)
- Howell, K.K., Lynch, M.E., Platzman, K.A., Smith,
G.H., Coles, C.D., Prenatal alcohol exposure
and ability, academic achievement, and school
functioning in adolescence A longitudinal
follow-up. Journal of Pediatric Psychology, 2006.
31 p. 116-126. - Kable, J.A., Coles, C. D., The impact of
prenatal alcohol on preschool academic
functioning. Poster presented at Society for
Research in Child Development (SRCD), Tampa, FL.,
2003, April. - Rasmussen, C. Bisanz, J. (2006). Mathematics
and working memory development in children with
Fetal Alcohol Spectrum Disorder. Alcoholism
Clinical and Experimental Research, 30, 231A - Kable, J. A., Coles, C. D., Taddeo, E. (in
press). Socio-cognitive habilitation using the
Math Interactive Learning Experience (MILE)
program for alcohol-affected children.
Alcoholism Clinical and Experimental Research.
40
445) General Strategies for Teaching Children with
FASD
- Preparing to Teach Students with FASD
- Children with FAS/FAE have 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 interests,
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 child 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 determine what is and is not
working.
41
45- 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 (e.g. during
morning routines, recess, lunchtime, subject
lessons, etc.) 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
42
46- 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 the child knows what is
coming next and what is expected - for example