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

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


1
Mathematics Instruction for Children with Fetal
Alcohol Spectrum Disorders A 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
3
Chapter 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

4
1) 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
10
2) 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
22
3) 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.

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  • Where Next?1
  • Once students form a solid knowledge base of
    numbers and counting, lessons may be advanced to
    actual computation in the horizontal and vertical
    forms. When a student is learning these
    procedures, it is important that they receive
    consistent help from teachers, aides, and
    parents. The same language, cues, and steps
    should be used so that the student does not
    become confused. However, it is also important
    to teach students a variety of techniques to
    solve these problems, particularly ones which
    will help the student learn more about number
    structure and composition.
  • It has been shown that adults rely more on
    addition and subtraction in every day life than
    multiplication and 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.

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  • 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

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  • The use of mnemonics may be useful to teach
    students a particular strategy. For example,
    RAVE CCC
  • R read carefully
  • A attend to key information that gives clues
    about necessary procedures
  • V visualize the problem
  • E estimate a potential solution
  • Once these steps have been taken, CCC outlines
    what should follow
  • C choose numbers
  • C calculate a solution
  • C check this solution (cross reference with
    your 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.

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  • Other techniques teachers may use to facilitate
    problem-solving competence in students with
    learning difficulties include
  • teaching difficult vocabulary before-hand

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  • 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.

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  • 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.

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  • 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.

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4) 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.

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  • In a study of 512 mother-child dyads,
    Goldschmidt6 examined the relation between
    maternal report of alcohol use during pregnancy
    and academic achievement of offspring at 6 years
    of age.
  • The authors found that drinking during the second
    trimester was related to difficulties in reading,
    spelling, and arithmetic. Furthermore, after
    controlling for IQ, prenatal alcohol exposure was
    still significantly related to arithmetic but
    only marginally related to reading and spelling.
    This indicates that these substantial deficits in
    arithmetic can not be solely attributed to a low
    IQ.
  • Others have found that 7-year-olds with prenatal
    alcohol 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.

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

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

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  • Preschool Children with Prenatal Alcohol Exposure
  • Little research has been conducted on math
    abilities in preschool children prenatally
    exposed to alcohol.
  • Kable and Coles15 looked at the relation between
    prenatal alcohol exposure and math and reading in
    4-year-old children from a high-risk (high
    alcohol exposure) and low-risk (low alcohol
    exposure) 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.

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  • Conclusions
  • There is considerable evidence indicating that
    children and adolescents with FASD and prenatal
    alcohol exposure have specific deficits in
    mathematics and particularly arithmetic.
  • These findings have been consistent across a
    multitude of both longitudinal studies and group
    comparison studies, even after controlling for
    many confounding variables and IQ. Thus, these
    math deficits are not simply due to a lower IQ
    among those with FASD, but rather prenatal
    alcohol 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

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  • 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).

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  • 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.

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5) 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.

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  • Kalberg and Buckley2 suggest that when developing
    an Individualized Program Plan (IPP) for a child
    with FASD it is important to also evaluate each
    childs current skill level and his or her
    specific academic needs.
  • Functional classroom assessments may also be
    useful to understand the childs real life
    abilities. The authors suggest observing each
    child in different natural settings (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

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  • Specific Classroom Interventions
  • Kalberg and Buckley2 also suggest some specific
    classroom interventions for children with FASD
  • 1) Structure and Systematic Teaching
  • structure environment and teaching and teach
    functional routines so the child knows what is
    coming next and what is expected
  • for example
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