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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

- 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

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

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

- 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

- 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

- 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

- 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

- 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

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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- Other techniques teachers may use to facilitate

problem-solving competence in students with

learning difficulties include - teaching difficult vocabulary before-hand

29

- 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

- 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

- 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

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.

33

- 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

- 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

- 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

- 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

- 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

- 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

- 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

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.

41

- 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

- 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