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鄧仲儀博士

Dr. Joey Tang

Mathematical Difficulties: Identification and Intervention in Early Primary School Years

¡ Why are some students poor at mathematics?數學困難的成因

¡ What is developmental dyscalculia?什麼是發展性數學障礙﹖

¡ Do all students who do badly at school mathematics have dyscalculia?數學成績偏低是否等同數學障礙﹖

¡ How do we identify students with dyscalculia?如何能識辨有數學障礙的學生?

¡ Weaknesses in cognitive processes, e.g.,

§ Symbolic understanding (數字理解)

§ Organization skills (組織能力)

§ Memory problems (記憶問題)

¡ Poor concentration span (缺乏專注力)

¡ Poor motivation (缺乏學習動機)

¡ Inefficient learning style (效率低的學習模式)

¡ Passivity (被動)

¡ Negativity (悲觀)

¡ Anxiety (焦慮)

¡ For dyscalculic probands, § 58% of monozygotic co-twins (同卵雙胞胎)

§ 39% of dizygotic co-twins (異卵雙胞胎 )were also dyscalculic (Alarcon et al., 1997)

¡ In a family study, approximately half of all siblings of children with dyscalculia were also dyscalculic, with a risk five to ten times greater than for the general population (Shalev & Gross-Tsur, 2001)

¡ A neuroimaging study of adolescent children who had been born preterm at 30 weeks gestation or less (Isaacs et al., 2001)

¡ An area in the left parietal lobe (左頂葉) where children without a deficit in calculationability have more grey matter than those who do have this deficit

¡ A structural neural correlate of calculation ability in a group of neurologically normal individuals

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¡ The first systematic study of specific deficits in learning about numbers and arithmetic was published by Czechoslovakian psychologist LadislavKosc (1974), who introduced the term

“Developmental Dyscalculia” 發展性數學障礙

¡ “Arithmetic learning disabilities”(Koontz & Berch, 1996)

¡ “Specific arithmetic difficulties”(Lewis, Hitch, & Walker, 1994)

¡ “Specific arithmetic learning difficulties”(McLean & Hitch, 1999)

¡ “Mathematical disability” (Geary, 1993)

¡ “Psychological difficulties in mathematics” (Allardice & Ginsburg, 1983)

¡ Differences in terminology have led to differences in criteria for assigning children to the category

¡ Traditional definitions (e.g., DSM-IV) are somewhat arbitrary

¡ E.g., DSM-IV (精神疾病診斷準則手冊 — 第四版)

¡ 精神疾病診斷準則手冊 — 第四版

¡ Mathematics Disorder 數學疾患

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¡ The approach taken by the U.K. Department for Education and Skills

¡ It defines dyscalculia in terms of the core characteristics as discovered empirically

¡ 英國教育局採用的數學障礙定義是根據學術研究結果訂出

數學障礙是一種影響學生學習算術的症狀。有數學障礙的學生通常對理解數字概念感到困難 ，缺乏對數字掌握的直覺，以及對運算公式和步驟感到困難。他們對運算欠缺信心 ， 因此可能只是靠機械化的方法來計出正確答案 。

¡ The “defective number module hypothesis” (Butterworth, 2005)

數學單元缺陷論

¡ The cognitive deficit in Developmental Dyscalculia (DD) is an inability to deal with exact numerosities

數學障礙是在絕對(可數)數量上缺乏認知

¡ Age range: 6 - 14 years¡ Administration: Individual ¡ Testing time: 15-30 minutes

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¡ Simple reaction time task (反應測試項目)

¡ Capacity subscales (基本能力測試)§ Dot enumeration task (點和數的配對項目)§ Number comparison task (數字比較項目)

¡ Achievement subscales (成績測試)§ Addition task (加數項目)§ Multiplication task (乘數項目)

¡ Nationally standardised全國 [英國] 標準化

¡ National average = 100, SD = 15

¡ Quick screening

¡ Suitable for teachers and psychologists

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¡ Tested a group of 9-year-old children with dyscalculia

Dot counting

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Charles¡ 30 years old¡ Adult dyscalculic¡ He always uses his

fingers to count

Dot enumeration

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¡ Number comparison task

Controls Charles

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¡ Teachers were asked to nominate children who they felt were of average general ability but had serious difficulties during numeracy lessons

¡ For each of these children, a control child of average mathematical ability was tested who was of the same gender and from the same class in school

¡ 36 children (8- to 9-year-olds)

¡ < 81 on at least one of the two tasks of the “capacity subscale” (基本能力測試) of the Dyscalculia Screener (test average of the nationally standardised score = 100, SD = 15)

¡ IQ score within the normal range

¡ Reading score within the normal range

¡ < 81 on the “achievement subscale” of the Dyscalculia Screener

¡ Within the normal range (> 89) in both tasks of the “capacity subscale”

¡ IQ score within the normal range

¡ Two children classified as having DD§ DD1 failed the symbolic number comparison task§ DD2 failed the dot enumeration task

¡ Eleven children classified as having LN§ Poor performance in the addition task

¡ Failed symbolic number comparison¡ Normal non-symbolic comparison § Approximate comparison § Area and numerosity comparison

¡ Poor exact (symbolic) addition¡ But normal approximate addition

¡ His disability cannot be interpreted in terms of a deficit in the approximate system

¡ A normal non-symbolic exact numerosity system

¡ His overall performance is similar to that of the LN group

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¡ Failed dot enumeration¡ Poor non-symbolic numerosity comparison¡ Normal area comparisonàA deep-rooted deficit in the capacity to represent

exact numerosities

¡ Abnormally poor on addition¡ Normal approximate addition and subtraction

¡ His DD condition cannot be explained by a deficit in the approximate system

¡ Low numeracy without developmental dyscalculia

¡ Normal on all non-symbolic tasksà Intact basic numerical concepts

¡ Significant impairment on exact (symbolic) addition

¡ Intervention for these children should stress the linkage of symbols to concepts

¡ Three-day old chicks can do this!

Arithmetic in Newborn Chickshttp://www.number-sense.co.uk/chicks/

0.000.100.200.300.400.500.600.700.800.90

S1 S2 S3 S4 S5

Average chick performance

Age of participants: 11-12 years

Accuracy

Performance of Children with Low Numeracy

¡ Development of a web-based testing platform (網上測試平台) modelled on Butterworth’s (2003) Dyscalculia Screener

Current Study目前的研究工作

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¡ Simple reaction time task (反應測試項目)

¡ Capacity subscales (基本能力測試)§ Number comparison task (數字比較項目)§ Dot-number matching task (點和數的配對項目)

¡ Achievement subscales (成績測試)§ Addition task (加數項目)§ Subtraction task (減數項目)§ Multiplication task (乘數項目)

Screening Tasks網上識別項目

Number Comparison Task 數字比較項目

Dot-number Matching Task 點和數的配對項目

Dot-number Matching Task 點和數的配對項目

Addition task 加數項目

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Subtraction Task 減數項目

¡ Participants: 180 right-handed university students (with equal gender ratio; mean age = 20.1 years, SD=1.2, range=18-25) born in Hong Kong, with Cantonese as both their mother tongue and dominant language

¡ On average, participants’ percentage accuracy was high (all mean >90%; accuracy threshold = 70%).

TaskCronbach’s

AlphaNumber of Trials

Number Comparison .977 32Dot-number Matching .920 32Addition .913 16Subtraction .922 16Multiplication .888 16Simple Reaction (Left Hand) .901 16Simple Reaction (Right Hand) .928 16

Study 1: Internal Consistency內在一致性

¡ Participants: right-handed girls from a single-sexed school, born in Hong Kong, with Cantonese as both their mother tongue and dominant language, and had an IQ score of at least 85 on the Raven’s Standard Progressive Matrices

¡ Accuracy threshold was set at 70% for fifth graders, 60% for fourth graders, and 50% for children in grades 1 to 3

¡ Data from 381 participants, who fit the above criteria, were analysed

¡

Study 2: Cross-sectional Study橫斷面研究

¡ All analyses were carried out on mean of median adjusted inverse efficiency (AIE), except for the simple reaction time tasks which used mean of median reaction time (RT)

AIE =

Mean of median reaction time for correct responses -Mean of median simple reaction time

Proportion of correct responses

Dependent Variable依變項

¡ A significant main effect of grade was observed in every task (all p <.001)

Grade Differences

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Results研究結果

¡ Participants: 248 children from different schools, selected using the same criteria as in Study 2

Male Female TotalN Mean

Age (y:m)

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(y:m)

N Mean Age

(y:m)Lower Grades 45 7:1 34 7:3 79 7:2Middle Grades 52 9:1 47 9:0 99 9:0Upper Grades 29 10:10 41 11:0 70 10:11

Study 3: Cross-sectional Study橫斷面研究

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¡ A significant main effect of grade was observed in every task (all p≤.001)

¡ Significant gender differences were only observed in the simple reaction time tasks (both p≤.05), with girls responding slower than boys

¡ Grade was never observed to interact significantly with gender

Results研究結果

¡ Significant differences between adjacent grade groups in all tasks, except between middle and upper grades in the number comparison task

¡ As children move from lower to upper grades, they became more efficient in their responses

Grade Differences

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†Calculations were based on mean RT

Results研究結果

¡ Participants: 20 children (mean age= 10.2 years; SD=0.8; range=8-11) who lagged behind in mathematics achievement as reflected by a locally standardised mathematics test, the Learning Achievement Measurement Kit (LAMK) 2.0, were referred by clinical psychologists

¡ Typically-developing children were matched on age, gender, handedness, mother tongue, dominant language, and IQ to each proband

Study 4: Clinical Cases臨床個案

¡ Using the Bayesian Inferential methods (Crawford & Garthwaite, 2007), 15 children with the typical developmental dyscalculia (DD) profile (i.e., those who performed significantly below average on either of the capacity subscales) were identified

¡ All of these children also performed below average on at least one of the arithmetic subscales

Results研究結果

¡ Importantly, they were distinguished from three children who showed the low numeracy (LN) profile, i.e., those with below-average performance in at least one of the achievement subscales but nonetheless performed in the normal range on both of the capacity subscales

Results研究結果

¡ In addition, two children showed comparable performance in all of the capacity and arithmetic subscales

¡ An inspection of their clinical records indicates that both of them have dyslexia and/or language impairment

¡ Their poor score on LAMK 2.0 could therefore be at least partially attributed to the poor understanding of the written verbal calculation problems used in the test

Results研究結果

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¡ How does anxiety contribute to poor arithmetic?

¡ Can maths anxiety （數學焦慮）be separated from test anxiety （考試焦慮）?

Anxiety and Poor Arithmetic 焦慮與運算偏弱(Tang & Chan, in preparation)

Phase I¡ 199 university students ¡ Mathematics Anxiety Rating Scale (MARS)

(Suinn & Winston, 2003) ¡ Test Anxiety Inventory (TAI)

(Spielberger, 1980)

Methods研究方法

Phase II¡ 36 students from Phase I were divided into high,

medium, and low maths anxiety (MA) groups depending on their MARS scores

¡ Two paper-and-pencil arithmetic tests (timed and untimed)

¡ Two- and three-column addition and subtraction problems§ Simple problems: no carry or borrow operation§ Complex problems: a carry or borrow operation

Methods研究方法

Phase I¡ A significant correlation between MARS and TAI

scores (r = .451, p < .01), Phase II¡ ANOVA on mean error rates revealed a significant

group x problem complexity interaction (F(2,33) = 3.64, p = .037)

¡ Time pressure had a significant effect on arithmetic performance, but did not interact significantly with group

Results研究結果

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Results研究結果

¡ Re-running the ANOVA with test anxiety scores as the covariate

¡ The group x problem complexity interaction remained significant (F(2,32) = 3.33, p = .049)

Results研究結果

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¡ Time pressure affects arithmetic performance¡ But the effect is similar across individuals with

different levels of maths anxiety¡ Highly maths anxious individuals’ poor

arithmetic performance is limited to complex but not simple problems

¡ Observed not only in addition problems (with a carry operation) as previously reported, but also in subtraction problems (with a borrow operation)

Summary總結

¡ Our findings provide the first report of a detrimental maths anxiety effect on complex arithmetic performance after partialling out the effect of test anxiety

¡ Also demonstrate the unique contribution of maths anxiety to poor arithmetic performance

Summary總結

¡ The detrimental effect of maths anxiety to poor arithmetic performance may be explained by a smaller working memory span, or reduced attention due to intrusive thoughts and worries, or both

¡ The intrusive and worrying thoughts act like a secondary task, adding load to the already limited working memory in highly maths anxious individuals

¡ Thus, when they have to solve complex problems involving carrying or borrowing which rely on the use of working memory to keep track of the calculation procedures, their performance suffers

¡ Pupils who find mathematics particularly difficult – whether they are dyscalculic or not – are often anxious or distressed about mathematics

¡ It is not helpful to stress the importance of mathematics in daily life or in academic advancement as the pupils already know this

¡ Stressing the importance of maths will only increase anxiety and distress

¡ Learning is more effective when it is enjoyable and relaxed

¡ Be patient

¡ Be supportive

¡ There are many causes of low mathematical achievement in addition to dyscalculia

¡ The first task is to discover why the pupil had a low score on a maths achievement test

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¡ The pupil may not have been trying on this occasion

¡ This can be established by asking the pupil to re-take the test

¡ Alternatively, another test of achievement can be used

¡ Pupils who fall behind, for whatever reason, may find it very hard to catch up

¡ The gap between what is expected of them in class and what they are competent to do will grow wider and wider unless steps are taken to help the pupil make up lost ground

¡ Intervention: back to basics

¡ Number concepts¡ Base 10 and place value¡ Addition and subtraction combinations

¡ Find alternative perspectives¡ E.g., use pictorial representations

¡ Dyscalculia Guidance

¡ Brian Butterworth & Dorian Yeo

¡ Dyscalculia Toolkit

¡ Ronit Bird

¡ Dyscalculic students’ underlying problem is in understanding numerosities

¡ Interventions should stress this very basic aspect of arithmetic

¡ The use of sets of objects for counting and manipulation may help to ground concepts of numerosity

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¡ It will not help a pupil in primary school to practise number bonds and tables until the numerosity concept is firmly established

¡ Attempting to induce rote learning of number bonds and tables could lead to frustration and avoidance

How many dots?

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A Bad Example of Learning Technology

¡ Confusing interface¡ Random generation of task level¡ Conflict between answer and learner’s action¡ Useless extrinsic feedback

Problems

¡ Clear and meaningful goal¡ Opportunity to act or construct an answer¡ Comparison between action and goal¡ Useful feedback¡ Chance to modify and improve¡ Adaptive and independent learning

What Constitutes a Good Program

Butterworth, B., & Laurillard, D. (2010)

An Adaptive Program for Dot-number Matching

http://www.number-sense.co.uk/numberbonds/

An Adaptive Program for Number Bonds

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¡ Butterworth, B. (2003). Dyscalculia Screener. London: nferNelson Publishing Company Ltd.

¡ Butterworth, B. (2005). Developmental dyscalculia. In Campbell, J. I. D. (ed.): Handbook of Mathematical Cognition. Hove: Psychology Press, 455-467.

¡ Butterworth, B. (2011). Foundational numerical capacities and the origins of dyscalculia. Trends in Cognitive Sciences, 14 (12), 534–541.

¡ Butterworth, B., & Laurillard, D. (2010). Low numeracy and dyscalculia: Identification and intervention. ZDM Mathematics Education, Special Issue on Cognitive Neuroscience and Mathematics Learning, 42(6), 527-539.

¡ Butterworth, B., Varma, S., & Laurillard, D. (2011). Dyscalculia: From brain to education. Science, 332, 1049-1053.

¡ Iuculano, T., Tang, J., Hall, C., & Butterworth, B. (2008). Core information processing deficits in developmental dyscalculia and low numeracy. Developmental Science, 11(5), 669-680.