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

1. Mathematics Instruction for Children with Fetal Alcohol Spectrum Disorder:A Handbook for Educators Carmen Rasmussen, PhD Katy Wyper, BSc Department of Pediatrics University of Alberta

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@ualberta.ca

3. Chapter Overview • Mathematics Deficits in Children with FASD (p. 1) • Children with PAE • Adolescents with PAE • Preschool Children with PAE • Conclusions • General Strategies for Teaching Children with FASD (p. 8) • Preparing to Teach Students with FASD • Specific Classroom Interventions • Helpful Educational Strategies • Behavioral Interventions • Stages of Math Development (p. 18) • Learning Framework in Number • Part A: Early Arithmetic Strategies & Base-Ten Arithmetic Strategies • Part B: Forward Number Word Sequences, Backward Number Word Sequences, & Numerical Identification • Part C: Other Aspects of Early Arithmetic • Strategy Competence • Characteristics of Students with Math Difficulties (p. 24) • Overview • Math for Students with Disabilities • Assessment of Math Difficulties • Language Ability and Math Difficulties • Strategies for Teaching Children with Math Difficulties (p. 36) • Goals of Instruction • Student Centered Approach • General Considerations • Helpful Tips • Teaching Problem-Solving • Strategies for Teaching Children with FASD Math (p. 49)

4. 1) Mathematics Deficits in Children with FASD Children with Prenatal Alcohol Exposure • The most direct evidence for the effect of prenatal alcohol exposure on mathematics difficulties among offspring comes from the landmark longitudinal study by Streissguth, Barr, Sampson, and Bookstein1. • Over 500 parent-child dyads participants, with about 250 of the mothers classified as heavier drinkers and about 250 as infrequent drinkers or as abstaining from alcohol (based on maternal report of alcohol use during mid-pregnancy). • From preschool to adolescence, these children were tested on a variety of outcome variables including IQ, academic achievement, neurobehavioral ratings, cognitive and memory measures, and teacher ratings. • Of all these outcome variables, performance on arithmetic were the most highly correlated with prenatal alcohol exposure at age 42, 7 years3, 114, and 145. Thus, the more alcohol these children were exposed to, the poorer they did on tests arithmetic, and this relation with alcohol exposure was the strongest of all of the variables measured. • Furthermore, 91% of the children who performed poorly on arithmetic at age 7 were still low at age 14, highlighting the stability and robustness of this finding. For older children maternal binge drinking appeared to be most related to lower arithmetic performance. • The authors Streissguth5 highlighted the recurrent finding that arithmetic is especially difficult for individuals who were prenatally exposed to alcohol. 1

5. In a study of 512 mother-child dyads, Goldschmidt6 examined the relation between maternal report of alcohol use during pregnancy and academic achievement of offspring at 6 years of age. • The authors found that drinking during the second trimester was related to difficulties in reading, spelling, and arithmetic. Furthermore, after controlling for IQ, prenatal alcohol exposure was still significantly related to arithmetic but only marginally related to reading and spelling. This indicates that these substantial deficits in arithmetic can not be solely attributed to a low IQ. • Others have found that 7-year-olds with prenatal alcohol exposures have a slower processing speed and a specific deficit in processing numbers.7 • Furthermore, arithmetic is one of the only measures that differentiates children with FAS/FAE from those with ADHD, in that only those with FAS/FAE show deficits in arithmetic.8 • In another study, Coles9 examined the cognitive and academic abilities of children aged 5 to 9 years from three groups: a control group not exposed to alcohol; a group whose mothers stopped drinking during the second trimester; and group whose mothers drank throughout the pregnancy. • Of all the achievement subtests, math was the lowest score among both the alcohol exposed groups, but not the control group. 2

6. 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. 3

7. 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. 4

8. Preschool Children with Prenatal Alcohol Exposure • Little research has been conducted on math abilities in preschool children prenatally exposed to alcohol. • Kable and Coles15 looked at the relation between prenatal alcohol exposure and math and reading in 4-year-old children from a high-risk (high alcohol exposure) and low-risk (low alcohol exposure) groups and found that the high-risk group performed significantly lower than the low risk-group on math but not reading. • In a recent study, Rasmussen & Bisanz,16 examined the relation between mathematics and working memory in young children (aged 4 to 6 years of age) diagnosed with an FASD. • Children with FASD displayed significant difficulties on the two mathematics subtests (applied problems and quantitative concepts) which measure problem solving, and knowledge of math terms, concepts, symbols, number patterns, and sequences. • Age was negatively correlated with performance on the quantitative concepts subtest, indicating that older children performed worse, relative to the norm, than younger children on the quantitative concepts subtest. Thus quantitative concepts appear to be particularly difficult with age among children with FASD. • Moreover, children with FASD performed well below the norm on measures of working memory, which were correlated with math performance indicating that the math difficulties in children with FASD may result from underlying deficits in working memory. 5

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

10. References • Streissguth, Barr, Sampson, and Bookstein (1994) • Streissguth, 1989 • Streissguth, 1990 • Olson, 1992 • Streissguth, 1994 • Goldschmidt (1996) • Burden (2005) • (Coles 1997 • Coles 1991 • Streissguth et al 1991 • Kerns, 1997 • Kopera-Frye, (1996 • Jacobson et al. (2003) • Howell, (2006 • Kable and Coles (2003, April • Rasmussen & Bisanz, 2007 • Kable, (in press • Kable et al., in press 7

11. 2) General Strategies for Teaching Children with FASD Preparing to Teach Students with FASD • Children with FAS/FAE difficulties in social emotional, physical, and cognitive functioning (particularly learning, attention sequencing, memory, case and effect reasoning, and generalizations).1 • Some suggestions for preparing to teach children with FAS/FAE include:1 • Collect information to understand the student’s strengths and weaknesses. • look at the student’s history, previous report cards, psychological reports, IPPs, as well as family and medical background • talk with the child about their interested, concerns, and supports • talk with the parents about the child’s strengths and weaknesses • observe the child in the classroom to evaluate needs and strategies for support • Make a plan to determine what the child’s needs to be successful. • look at resources, manuals, handbooks • consult with other teachers and special education teachers, professionals, counsellors, and psychologists. • develop activities to focus on the most important needs of the child • Evaluate the plan to determined what is and is not working. • If the child is till having difficulties may need to make a referral for assistance (classroom aide). 8

12. Kalberg and Buckley2 suggest that when developing an Individualized Program Plan (IPP) for a child with FASD it is important to also evaluate each child’s current skill level and his or her specific academic needs. • Functional classroom assessments may also be useful to understand the child’s real life abilities. The authors suggest observing each child in different natural settings on a few different occasions to understand conditions that both disrupt and enhance each child’s functioning. • Important characteristics to observe: • skills • attention • independence • social interactions • language • strengths and interests • behavior 9

13. Specific Classroom Interventions • Kalberg and Buckley2 also suggest some specific classroom interventions for children with FASD: 1) Structure and Systematic Teaching • structure environment and teaching and teach functional routines so child knows what is coming next and what is expected 2) Visual Structure • individualized visual schedules, routines, visual organizations • visual instructions and visual cues • color coding, labelling areas of classroom and tasks • highlight important information on a task • ensures the environment and tasks are clear and predicable and helps with child with sequencing events, transitions, anticipating what will come next 3) Environmental Structure • keep environment simple with few distractions • have obviously defined work areas 4) Task Structure • structure tasks so that a child understands the task expectations, what steps to do, and what to do when finished 5) Cognitive Control Therapy • help each child to understand their own learning style and learning difficulties 6) Involve the Family • listen to the families desires and wants 10

14. The acronym ‘SCORES’ has been used to depict characteristics of a good classroom environment for students with an FASD:3 • S – Supervision, Structure, Simplicity • C – Communication, Consistency • O – Organizations • R – Rules (simple and concrete) • E – Expectations (realistic and attainable) • S – Self-esteem (acceptance and encouragement) • Finally Danna Ormstup4 suggested the following acronym to make your classroom ‘ROCK’ for a child with FASD: • R – Routine • O – Organized • C – Consistency • K – Knowledge base (know each child’s strengths and weaknesses) 11

15. Helpful Educational Strategies • Wescott5provides general strategies for educating children with FAS and FAE: • slow down and simply information • structure, use consistent daily routines • don’t overload them with stimuli • keep transitions constant • reduce words and verbal cues • focus on real life skills • set up simulated stores, banks, etc in the classroom • use metaphors with concrete choices • use visual cues (pictures, cartoons) to depict daily activities • promote good coordination and communication between parent and teacher • Kvigne et al.6 suggest many helpful education techniques for children with FAS/FAE including: • have a calm and quiet environment, using calm colors • minimize distractions and objects hanging on walls • use structure and routines, with simple and consistent rules • headphones may assist with quiet activities • help child prepare for transitions, give breaks throughout the day • use teaching methods that stimulate all senses • use concrete examples and picture calenderers • give children choices • always have the same activity in the same place 12

16. Burgess and Streissguth7 describe children with FAS/FAE as impulsive, having difficulty with transitions, poor judgment, not understanding consequences, and poor communication. They suggest guiding educational principles such as early intervention, focus on communication skills and making choices, and teaching social skills. To manage problematic behaviors the authors suggest teaching communication skills and making choices, planning ahead, and creating a balance between structure and independence. • Other general teaching strategies for children with FASD include:1 • organizing the classroom with few distractions • color code student’s material in one binder • use pictures • have quite work areas • use moderate lighting and heating and warm colors • have structure and consistent rules • prepare the child a head of time for changes and transitions • have simple rules with consistent consequences • avoid too many choices • don’t give too much homework • find a medium between not expecting too little and expecting too much • Sobsy8provide a list of instructions tips for children prenatally exposed to drugs and alcohol which include: • teach within the child’s cognitive and social aptitudes • teach towards each child’s unique learning style and strengths, and what they are able to do • use small steps and few words • use hands on materials and concrete examples, encourage participation in activities • use observation learning techniques, imitation, and physical aids • practice, and give feedback 13

17. Evensen9 suggests 12 important elements for success when teaching students with FASD: • encourage success for children with FASD • interact with the child’s family and respect their emotions • try a different approach when things are not working • structure • observe behaviour • interpret behaviour • ensure the environment is meeting the sensory needs of the child • use concrete language • know the memory difficulties faced by children with FASD • recognize their social and academics difficulties • appreciate the life transitions for individuals with FASD • reinforce and praise success 14

18. Behavioral Interventions • McLaughlin, Williams, and Howard10 review classroom behavioral interventions for children prenatally exposed to alcohol and drugs which include: • Contingency management • Token reinforcement programs: reinforce positive behaviors with tokens, either concrete (chips) or symbolic (points). Use tokens with social praise. • Contingency contracting: write out contingencies with behaviors and consequences on a contract for both the child and educator to sign. • Daily report cards: are also helpful when combined with a home reinforcement system. • Behavioral Self-Management • Self monitoring: in which the child does self-assessment and self-recording. This can improve attention, task completion, and reduce problematic behaviors • Self-instructional training: self-instruction to improve on-task behavior. For example, a teacher demonstrates a problem and solution, and then the student performs the task while saying the steps out loud, then whispering, then silently. • Self-managed drill and practice: Cover, Copy, and Compare technique,11 Student looks at problem, covers problem, answers problem, uncovers problem, and compares their response to the original. This technique is private and allows the child to work at his or her own pace. It is effective with children with behavior problems and disabilities, and has been successfully used in math. 15

19. Peer Tutoring • Classwide peer tutoring: helpful for students with poor achievement or who have disadvantages pr disabilities. • Reciprocal peer tutoring: students service as both tutor and tutee • Cross-age peer tutoring: having older student assist younger students • Direct Instruction • Direct instruction involves numerous student-teacher interactions, well-planned and sequenced lessons, and modern learning techniques.12 The aim is to teach more in less time by teaching in small groups in a fast pace, using a few examples that can be applied to a number of different situations, and giving instant and positive corrections.10 • McLaughlin et al.10 also suggest that because medications can be effective with student with ADHD they may be useful when combined with behavioral interventions for children with prenatal exposure to alcohol and drugs, but more research is needed on these effects. 16

20. References • Manitoba Manual • Kahlberg and Buckley (2007) • C and O Manual • Danna Ormstup (March, 2007) • Wescott (1991) • Kvigne, Struck, Engelhart and West • Burgess and Streissguth (1992) • Sobsy • Deb Evensen, (2007, March) • McLaughlin, Williams, and Howard (1998) • McLaughlin and Skinner (1996) • Engelmann and Carnine (1998) 7

21. 3) Stages of Math Development • According to the UK National Numeracy standards, by the end of the first year of formal math education, children should be able to:1 • 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 children’s 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 18

22. Part A • Early Arithmetical Strategies: • Emergent Counting – children are unable to count visible objects due to either not knowing words for numbers or not being able to coordinate the words with the objects. • Perceptual Counting – children are able to count perceived (ie. heard, seen, or felt) objects, but not objects in a screened collection • Figurative Counting – children can count objects in a screened collection but this counting is still rudimentary (e.g. when asked to add two collections and told how many object are in each, children count objects one by one instead of counting on from the largest screen.) • Initial Number Sequence – children are now able to “count-on” and to solve addition problems with one number missing (e.g. 4 + _ = 7). Children can also use some “count-down” strategies (e.g. 15 – 4 as 14, 13, 12, 11). • Intermediate Number Sequence – children are able to use count-down strategies more efficiently. • Facile Number Sequence – children can now use a range of strategies not limited to counting by ones (e.g. recognizing that there is a 10 in all teen numbers). • Once children have advanced to Stage 6, they progress through 3 levels involving the use of base-ten strategies. • Base-Ten Arithmetical Strategies: • Level 1 – Initial Concept of Ten – Children can count to and from 10 by ones but do not recognize ten as a unit. • Level 2 – Intermediate Concept of Ten – Children now recognize 10 as a unit, but cannot perform any operations on it without the components being represented in groups of ones (e.g. two open hands); they cannot perform operations on tens in the written form. • Level 3 – Facile Concept of Ten – Children are now able to solve addition and subtraction problems without material representations. 19

23. Part B • FNWS, BNWS, and Numeral Identification: • Number words are the spoken and heard names of numbers (Wright, et al). The LFIN draws an important distinction between a child being able to actually count and being able to recite a list numbers in the correct order. Knowledge of forward and backward number order sequences is a child’s ability to count a sequence of number words forwards and backwards, not only by ones but by other units as well. • Johansson2 suggests that children’s knowledge of number words is related to other numerical abilities. For example, children may recognize a structure in number word sequences and use this structure to solve arithmetic problems. There are three levels a child goes through to when learning how to arithmetics: • the child uses physical objects to represent addends (e.g. David has 3 apples and Simon has 2 apples. How many apples are there?) • the child uses non-physical representations to solve problems (e.g. verbal unit items) • the child uses known facts or procedures to solve problems • Numerals are the written and read form of numbers. Numeral identification is a child’s ability to produce the name of a given numeral. Identification is different from recognition in that to recognize, a child must simply pick out a named numeral among a random set as opposed to producing the name him or herself. 20

24. Part C • Other Aspects of Early Arithmetical Learning: • These aspects are not as directly addressed by the LFIN but are nevertheless related to components of parts A and B. • Combining and Partitioning: Children may learn to recognize combinations and partitions of numbers (e.g. one and four is five; seven is three and four). These sets of numbers become automotized so that children have knowledge of them without having to count one by one. • Spatial Patterns and Subitizing: This aspect involves a child’s ability to recognize spatial patterns such as dominos patterns, playing card patterns, or dot cards. To “subitize” is a technical psychological term which means to capture the number of dots in a stimulus without actually counting them. • TemporalSequences: These are stimuli, such as sounds or movements, that occur sequentially time. • Finger Patterns: Children’s use of fingers strategies increases in complexity as they advance through the stages of SEAL. Eventually it is expected that children will no longer rely on their fingers, but these strategies play a very important role in early stages. • Base-Five (Quinary-Based) Strategies: Base-five strategies are useful in situations that involve sets of five items. 21

25. Strategy Competence • In a study of children with math and reading difficulties, Torbeyns et al.3 concluded that strategy competence develops along the following four dimensions: • strategy repertoire • strategy distribution • strategy efficiency • strategy selection • Compared with typically developing children, children who have mathematical disabilities in the first and second grades: • have the same strategy repertoire (retrieval, counting) • use retrieval less • use more immature forms of counting • are slower at selecting strategies • implement strategies less accurately • make less adaptive strategy choices • Most of these differences between MD and typical children seem to decrease with age, however strategy frequency characteristics remain. Children with MD show less strategy development than typical children (e.g. they continue to rely on counting strategies, while typical children use retrieval at an increasing frequency) and these differences may exist as a result of a developmental delay instead of a developmental deficit. That is, the mathematical abilities of children with MD develop more slowly than those of typical children, but they eventually develop nonetheless. • Intervention should be directed towards the procedural skills that are lacking in children with MD. Further, it is possible that there are underlying cognitive factors, such as working memory, that contribute to the development of mathematical strategies. Finally, the children examined in this study had mathematical and reading disabilities, and the results therefore cannot be generalized to populations without the combination of these disabilities. 22

26. References • Wright, Martlund, & Stafford, 2000 • Johansson 2005 • Torbeyns et al. 23

27. 4) Characteristics of Students with Math Difficulties Overview • According to Chiappe,1 math difficulties (MD) appear to be the consequence of a specific deficit rather than a general learning problem. If MD were a result of some general deficit, those children with problems in math would also experience problems in other areas, but this is not the case. Two factors that may be responsible for MD some children encounter are problems with number representation and the inability to process numerical stimuli. Longitudinal research provides support for the latter. • Studies have documented the existence of number representation and processing as early as infancy and early childhood. 1 Interruptions in the normal development of these processes may be the cause of math deficits found in older children. An improper representation of number can cause difficulties in counting, number sense, and discriminating quantities. For example, some children are able to count from one to five, but do now know whether 4 is greater than 2 or 2 is greater than 4.1 24

28. Children with learning problems have difficulties describing what they are thinking when they added numbers.2 However, they use strategies similar to those used by typical children when adding numbers (count-all, and count-on, with or without the use of physical objects). This suggests that, similar to typically developing children, children with learning problems do in fact acknowledge relationships between numbers instead of simply depending on rote memorization when performing addition problems. • One issue to be aware of is that sometimes students may provide a correct answer to a math problem by using the wrong strategy. It is important to keep this in mind, because it could easily go unnoticed in a classroom situation.2 It has been documented that sometimes children try to hide their hands while counting on their fingers. Due to the fact that students with learning problems may never pass the point of depending on physical objects to count, it is important to encourage the use of these objects when performing math problems. 2 25

29. Math for Students with Disabilities3 • Students that have difficulties with math in elementary school seem to have more problems retrieving number facts in higher grades. This difficulty perpetuates into upper level math such as algebra. • Counting strategies: • another difference that shows up between students with and without math difficulties is the complexity of their counting strategies • young students with math difficulties may use the same strategies as students without difficulties, but they tend to make more mistakes • the strategies that students use to count are a good predictor of how receptive they will be to traditional teaching techniques • Reading difficulties seem to exacerbate the problems that students encounter in mathematics. • One of the primary deficits in students with math difficulties is poor calculation fluency (recalling number facts quickly and relying on simple strategies). 26

30. Number sense: • Defined as: • fluency in estimating and judging magnitude • ability to recognize unreasonable results • flexibility when mentally computing • ability to move among different representations and to use the most appropriate representation • Two indicators of number sense in young children are counting ability and quantity discrimination. Quantity discrimination may be associated with informal math learning that occurs outside of the school setting, whereas counting may be more dependent on formal education • Number sense may be used to predict future performance in other areas of math, the first four of which are influenced by instruction: • quantity discrimination/magnitude comparison • missing number in a sequence • number identification • rapid naming • working memory • Early intervention should focus on: • improved calculation fluency and accuracy • improved counting strategies (more sophisticated and efficient) • beginnings of a number sense 27

31. Some suggestions for interventions include:3 • encouraging student to depend on their retrieval skills as opposed to counting • technologies that allow individualized practice • instruction focusing on strategy development and use • automatization of number facts and teaching “shortcuts” • improves both number sense and fluency • small group work that promotes familiarity and comfort with numbers • developing math vocabulary • structured peer work • using visuals and multiple representations • teaching strategies that could be used as a “hook” for problem-solving • acknowledging areas of weakness and allowing for more practice and time spent working in these areas • the transition from concrete to abstract math concepts is imperative in the development of calculation fluency 28

32. Assessment of Math Difficulties4 • Problems that students with special needs often encounter while learning math include: • inadequate or unsuitable instruction • curriculum that is too fast-paced • lack of structure which promotes discovery learning • teachers’ use of language that does not math students’ level of understanding • early use of abstract symbols • trouble reading math word problems (students with reading difficulties) • problems with basic math relationships which propagate into higher-level math • insufficient revision of early learned math concepts • In order to avoid simply “watering-down” the math curriculum for students with learning difficulties, is may be useful to incorporate math in other areas of learning such as social studies, sciences, reading, and writing. • The first step towards fostering a more solid understanding of math in students with difficulties is to determine what they already know, identify any holes that may exist, and formulate a plan to fill these holes. This may be done by constructing “mathematical skills inventories” which reflect the curriculum to be taught. Teachers may keep track of the types of mistakes students are making, and use these patterns to identify weaknesses. • Informal interviews between teacher and student may also be a useful technique to identify skills and weaknesses. Several areas that are important in problem-solving ability are: • identifying what the problem is asking • picking out the relevant details • choosing the appropriate procedure to solve the problem • estimating a solution • calculating the solution • checking the solution 29

33. Asking questions like “why did the student have trouble with this area?”, “would the use of concrete objects or other aids help the student solve this problem?” and “is the student able to explain to me what to do?” may help determine the extent of difficulty, and where exactly the misunderstanding occurs in the problem-solving process. • To build on a student’s existing knowledge, it must first be determine how much the student knows. Assessment can be broken down into three Levels: • Level 1: The student has trouble with basic number. First, examine the student’s vocabulary of number relationships and conservation of number. Assessment must then be done by examining each of the following items in order: • sort by a single attribute • sort by two attributes • create equal sets using one-to-one matching • count objects to ten, then twenty • recognize numerals to ten, then twenty • correctly order number symbols to ten, then twenty • write down spoken numbers to ten, then twenty • understand ordinality (first, seventh, fourth, etc) • add numbers below ten with counters and in writing • subtract numbers below ten with counters and in writing • count-on in addition • solve simple oral addition and subtraction problems (numbers below ten) • familiarity with coins and paper currency 30

34. Level 2: Performance is slightly higher than in Level 1. Assess the following: • mental addition below twenty • mental problem-solving without using fingers or tally-marking • mental subtraction; is there a discrepancy between addition and subtraction performance? • vertical and horizontal written addition • understanding of addition commutativity (i.e. the order of addends does not matter); does the student always count-on from the largest number? • understanding of additive composition (every possible way of producing a number – e.g. 4 is 1+3, 2+2, 3+1, and 4+0) • understanding of the complementary order of addition and subtraction problems. For example, 7 = 3 + 4; 3 + 4 = 7 and 5 – 3 = 2; 5 – 2 = 3. • translate an operation observed in concrete objects to a written equation • transfer a written equation into a concrete equation • translate a real-life scenario into a written problem and solve it • recognize and write numbers up to fifty • tell digital and analogue time • list the days of the week • list the months of the year 31

35. Level 3: The student is able to perform most of the item in Level 1 and 2: • read and write numbers to 100, then 1000 • read and write money additions • mentally compute halves or doubles • perform mental addition of money; determine amounts of change using count-on • memorize and recite multiplication tables • add hundreds, tens, units and thousands, hundreds, tens, units with and without carrying • know the place values with thousands, hundreds, tens, units • subtraction algorithm with and without exchanging columns • correctly perform the multiplication algorithm • correctly perform the division algorithm • understand fractions • correctly read and solve basic word problems • Translating abstract concepts into tangible, concrete problems is helpful for children with learning disabilities. It is important however, to ensure that students do not learn to rely on these physical objects, and that they gradually transition from concrete to abstract understanding. 32

36. Language Ability and Math Difficulties5 • Children with specific language impairment (SLI) appear to have difficulties in counting and knowledge of basic number facts, however they are quite successful on written calculations with small numbers. One area that may cause trouble for students with SLI is the increased amount and complexity of mathematical vocabulary these children are exposed to in higher elementary school (grades 4 and 5). This presents a problem because children with SLI have a tough time retrieving information that has been rote memorized. Another area in which children with SLI show difficulty is information-processing and this difficulty can produce challenges with the recall of declarative knowledge, and procedural knowledge. The mathematics required of upper elementary school students demands a combination of conceptual, procedural and declarative knowledge – all of which present problems for children with SLI. • Students with SLI are poorer at recalling number facts as well as using correct procedures for problem solving. They tend to rely more on simple strategies like counting and less on advanced strategies like retrieval. • Children with SLI perform better on written calculation tasks when they are un-timed, suggesting that these children are indeed capable of performing well, just at a slower pace than typically developing children. Written calculation task performance was much worse when children were timed. Tasks that are performed under a time constraint tend to load on working memory, which ties in to why children with SLI would show difficulties on such problems. 33

37. It is possible that the discrepancy between information-processing abilities in typically developing children and children with SLI may be explained in part by the improved automaticity in typically developing children. If true, children with SLI who are given the opportunity to practice may show improvements in their own automaticity, thus freeing up cognitive resources that could be used for other processes. Moreover, children’s performance on timed tasks should improve if they are taught strategies to automatize because they can spend less time tasks that were once controlled and consciously attended to. Two ways in which automatization might be encouraged are computer-based interventions and paper-and-pencil “drill and practice” games. • Another factor that may play role in the difficulty that children with SLI encounter when it comes to math problems is that many of these children are living in poverty and often receive poorer education that children from a more affluent family. • Children with SLI experienced may problems with the procedural aspect of calculations. The author suggests two ways to rectify this problem: (1) by encouraging students to “think through” the steps involved in answering a particular question, and (2) instructing children to ask themselves questions such as “what operation must I use for this problem?” Teaching students to confirm their answers to math problems (e.g. 87 – 24 = 63, 63 + 24 = 87) may help them understand mathematical concepts and relationships. • Finally, children’s attitudes and feelings towards math, and interactions with other students. 34

38. References • Chiappe • Hanrahan • Gersten et al • Chap 12 • Fazio 1999 35

39. 5) Strategies for Teaching Children with Math Difficulties Goals of Instruction1 • There are five goals of mathematics education: to learn the value of mathematics, to build confidence in mathematic ability, to learn how to solve mathematical problems, to learn how to communicate mathematically, and reason mathematically. • Students proficient in math possess the following skills: • Conceptual understanding: understanding of concepts, relations, and operations. • Procedural fluency: perform procedures with skill, speed, and accuracy. • Strategic competence: develop appropriate plans for problem-solving. • Adaptive reasoning: the ability to think about problems flexibly and from different perspectives. • Productive disposition: enjoying and appreciating math, and being motivated to improve mathematical ability. 36

40. It is important to distinguish between and identify math difficulties and disabilities, because the identification and intervention may prevent children with math weaknesses from developing a full disability. • New amendments to American legislation have recently been made in a project called IDEA. These modifications are geared towards helping children with learning disabilities as well as their families and teachers. Three areas that are affected by the amendments are criteria for determination of eligibility, whether the child will respond to research-based intervention, and the percentage of federal funds that may be allotted to early intervention services. • Distinguishing between students with learning difficulties and learning disabilities is also important in order to provide the most appropriate instruction. Children with learning disabilities have no trouble generalizing strategies to other related areas of learning; however children with learning difficulties have more problems making such generalization.2 37

41. Student-Centered Approach • It was once believed that math should be taught in the form of rule-based instruction, whereas now, research supports a more student-focused form of instruction. That is, teachers should consider students’ existing mathematical knowledge and provide an environment in which realistic problems combine with and strengthen this existing knowledge. This process is called Realistic Mathematics Education (RME).2 • According to Milo et al.2 one responsibility of the teacher is to facilitate knowledge construction based on the students’ existing knowledge. One kind of instruction is guiding instruction: • Guiding instruction: the instructor’s role is to guide the student to a more solid understanding of math by combining new knowledge with the student’s own contributions (guiding instruction) as opposed to simply directing the students about mathematical concepts (directing instruction). In guiding instruction, students are encouraged to reflect upon new strategies that they learn, which teaches them to choose more appropriate strategies in the futures. • However, students with special needs may not benefit from this type of instruction. Generally, students with learning problems have difficulties structuring the strategies that they learn. Consequently, a more directive instructional approach may be more appropriate: • Directing instruction: the teacher provides the student with explicit rules and structure may reduce the ambiguity that sometimes exists in guiding instruction. 38

42. In the directing instruction, one specific strategy may be taught in isolation, as opposed to guiding instruction, where students are encouraged to compare and choose (based on their own existing knowledge) among multiple strategies, and then to explain their choices. Typically-developing children may benefit most from guiding instruction, while children with special needs benefit more from directing instruction. The use of supporting models (e.g. number lines, number position schemes) also contribute to the special needs students’ understanding of appropriate and effective strategy use. • Children may tend to rely more on strategies formally learned in school and less on strategies they may have learned before entering school.3 Children also show overconfidence in these strategies, regardless of their effectiveness. Because school-taught strategies tend to be fairly rigid, it is important to emphasize flexibility. 39

43. General Considerations Some important points to remember when providing instruction:4 • Differentiation: recognize differences among individual students and modify instruction according to these differences. This method may be used with students who have disabilities or learning problems, and also those who are the most gifted. Examples: • personalized learning objectives for each student • adapting curricula to suit the students’ cognitive level • different paths of learning for different learning styles • spend more or less time on lessons depending on students’ rates of learning • modifying instructional resources (manuals, texts) • allow the students to produce work through a variety of media • be flexible with grouping students • adjusting the amount of help or guidance giving to each student • Simplicity: There are many different ways to “adjust,” “modify,” or “adapt” instruction. However, it is best to keep things simple. • use only one or two differentiation strategies in the classroom at once • only when necessary • be sure to return to your regular teaching style and curriculum as soon as possible • It is important to ensure that modifications are only temporary, as lower-achieving students will not benefit from constantly receiving lessons that are less challenging than higher-achieving students. Decreasing the demands placed on lower-ability students may further widen the gap between lower-achieving and higher-achieving students. 40

44. CARPET PATCH: A mnemonic device which summarizes methods that teachers may use to implement differentiation. • C – curriculum content • A – activities • R – resource materials • P – products from lessons (what students are asked to produce) • E – environment • T – teaching strategies • P – pace • A – amount of assistance • T – testing and grading • C – classroom groupings • H – homework assignments • Other helpful strategies: • re-teach some concepts using different language and examples • use different techniques to maintain interest of less motivated students • modify the amount and detail of feedback given to students • provide opportunity for extra practice for those students who need it • extension work for more able students • Students with disabilities: all of the strategies described above may be appropriate for students with disabilities. In addition, useful information may be accessed in the student’s individual education plan (IEP). • Explicit and direct instruction is often the most useful for students with learning disabilities or difficulties. Practical, hands-on activities, group work and verbal discussion about math are also important in the facilitation of math learning. Group work is most useful when each student has an opportunity to contribute. 41

45. Helpful Tips1 • Counting. Sometimes children will learn to memorize counting rhymes, but not connect these rhymes with the actual counting of physical objects. Guidance (hand-over-hand or direct, explicit teaching) may help students to make this connection, which is so fundamental in early math learning. • Numerals. Familiarity and recognition of numerals may be fostered by repetition presentation in the form of flash cards or other games. Over-learning gives lower-ability students the chance to establish a solid base on which they can build higher math skills. • Writtennumbers. Children with learning difficulties may have problems if introduced to written number symbols too early. A good alternative is to use dot schemes, tally marks, or other number representations before using number symbols. • Number Facts. Another area of weakness for some students with learning difficulties is the automatic retrieval of number facts (e.g. 4 + 2 = 6) as well as knowledge about mathematical procedures (what to do when you see ‘+’). Ensuring that students learn facts and computational procedures through increased regular practice and number games will allow them to solve math problems more quickly and easily. Calculators can also be used to aid students with computational difficulties, but some teachers may not wish to substitute traditional written math with an electronic device. • Number Games. Instead of having children complete traditional exercises and worksheets, turn math learning into a game. Using small candies or toys can make lessons interesting and fun, but it is important to make sure that these lessons remains educational, not just entertaining. 42

46. Where Next?1 • Once students form a solid knowledge base of numbers and counting, lessons may be advanced to actual computation in the horizontal and vertical forms. When a student is learning these procedures, it is important that they receive consistent help from teachers, aides, and parents. The same language, cues, and steps should be used so that the student does not become confused. However, it is also important to teach students a variety of techniques to solve these problems, particularly ones which will help the student learn more about number structure and composition. • It has been shown that adults rely more on addition and subtraction in every day life than multiplication and division, so if a teacher must prioritize math curriculum, it may be useful to focus most on addition and subtraction, followed by multiplication, and finally division. • Students with perceptual problems may require slight modifications in teaching material in order to perform on paper-and-pencil problems. Some examples that may be useful are thick vertical lines, squared paper, and small arrows or dots that the students may follow on the page. 43

47. Teaching Problem-Solving1 • The next step in math learning, problem-solving, could be a particularly difficult task for students with disabilities because they may have trouble in the following areas: • reading the words • understanding specific words within the problem • comprehending the problem in general • linking an appropriate strategy to the problem • Consequently, students may feel overwhelmed or hopeless when attempting such problems and it is important to teach them how to feel confident and comfortable working through these problems. • People generally problem-solve in the following order: • interpret the target problem • identify strategies needed to solve the problem • change the problem into an appropriate algorithm • perform computations • evaluate the solution • We also self-monitor and self-correct throughout the entire problem-solving process. Students with learning difficulties should be taught these skills through direct instruction and explanation early on in order to become independent problem-solvers later on. Several techniques teachers may use are: • modeling and demonstration of the appropriate strategies used to solving routine and non-routing problems • talk through the steps that should be taken, and questions that should be asked during the entire process • evaluate the steps and procedures used to solve the problem once it has been completed 44

48. The use of mnemonics may be useful to teach students a particular strategy. For example, RAVE CCC: • R – Read carefully • A – Attend to key information that gives clues about necessary procedures • V – Visualize the problem • E – Estimate a potential solution • Once these steps have been taken, CCC outlines what should follow: • C – Choose numbers • C – Calculate a solution • C – Check this solution (cross reference with your estimate) • Ideally, as students become more comfortable with problem-solving procedures and strategies, teachers may move from direct instruction to less-involved guided practice and eventually the student may become and independent problem-solver. • The use of calculations does not impede students’ progression from basic number sense, to computational skill, to problem-solving proficiency. In fact, the use of a calculator may allow teachers to focus more on teaching higher-level problem-solving strategies, and it has even been suggested that students who use calculators develop more positive feelings about math. 45

49. Other techniques teachers may use to facilitate problem-solving competence in students with learning difficulties include: • teaching difficult vocabulary before-hand • using cues to show students where to begin and where to go from there (e.g. arrows) • connecting problems with students’ own life • allowing students to create problems and have other people solve them • encouraging the use self-monitoring and self-correction • As always, teachers should consider each individual student’s existing knowledge when planning problem-solving lessons. • Intervention strategies that are aimed at a child’s particular difficulty would likely be most effective. Components of arithmetic that have been identified by teachers and researchers as particularly important are related to counting, the use of written symbols, place value and derived fact strategies, word problems, relations between concrete, verbal and numerical forms of problems, estimation, and remembering number facts. • In terms counting, young children most often encounter problems with regard to the order-irrelevance issue, and repeated addition and subtraction by one. Problems in these areas are improved by practicing counting and cardinality questions starting with very small numbers and working up. 46

50. Children’s understanding of written symbols can be solidified by having the child practice reading and writing simple arithmetic equations. Place value can be more clearly taught by presenting children with different forms of addition including written numbers, number lines and blocks, physical objects (hands, fingers, blocks), currency (pennies and dimes), and any kind of mathematical apparatus. To clarify children’s understanding of word problems a useful technique is to present addition and subtraction word problems, and discuss their characteristics with the child. • The relation between concrete, verbal, and numerical forms of arithmetic problems appears to cause particular difficulty among children. To resolve this difficulty, it has been found useful to present the similarities among different forms and demonstrate why each form has the same answer. • Derived fact strategies can be taught by presenting two similar arithmetic problems to children, teaching an effective strategy for solving one of the problems, and then explaining how and why the same strategy may be used for the second problem. Lessons on estimation are often successful when children are asked to judge estimates made by make-believe characters. That is, children are shown a group of arithmetic problems as well as proposed answers (given by pretend characters), and asked first to evaluate the answers and then provide a justification for their evaluation. Finally, memory for number facts can be improved by repeatedly presenting children with simple arithmetic facts (e.g. 2 + 2 = 4) over multiple sessions and playing games to strengthen memory for these facts. • Both teachers and students who have tested these intervention techniques deemed them useful and fun, and a valuable way to spend one-on-one time with each other. Further, a particularly meaningful outcome of these intervention strategies is that children often gained self-esteem and confidence in their mathematical abilities. 47