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Articles Of Research Identifying Abacus Value

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Old 11-05-2010, 03:56 PM   #1
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Articles Of Research Identifying Abacus Value



A Dissertation Submitted for the Dr. Degree in Abacus Study

Associate professor of University of Chicago, Dr.James Stigler, wrote his Doctoral Dissertation about abacus study. He stayed in Taiwan for two years and did research to the students. He gave a test before taking abacus lessons, then taking lessons and after that he completed abacus lessons. Then he wrote doctoral dissertation and received PhD. His major is in Psychology in Human Behavior. Later, he wrote many theses, including the "Mental Abacus - The Effect of Abacus Training on Chinese Children's Mental Calculation"., Dr.James Stigler, "Mental Abacus - The Effect of Abacus Training on Chinese Children's Mental Calculation".
Evaluation of Memory in Abacus Learners.

Published in Indian J Physiol Pharmacol. 2006 Jul-Sep;50(3):225-33)

Department of Physiology, Stanley Medical College and Hospital, Chennai 600 003.

Abacus is a method used by Chinese, Japanese and Koreans to improve mathematical skills. This system has now invaded our country. The improvement in mathematical skills is said to be due to a coordinated functioning of both right and left hemisphere. As learning and memory in any field is achieved by coordinating and analyzing the different sensory inputs, whether an abacus trainee would also improve the short-term memory as a whole was evaluated in our study. 50 children of average IQ between 5 and 12 years from 2 regular schools and 50 from an abacus institute were evaluated for short-term memory before and after a period of one and two years. The memory tests were taken from Wechsler memory scale, Mini mental state examination, Mann - Buitar visual memory screen for objects. The results showed that the abacus learners at the end of one and two years had a better visual and auditory memory when compared to non-abacus learners.

About skilled Abacus Users and the training of intelligence.
Mr. Lin and Mr. Haqits, professors in Osaka Education University, studied abacus users and found that abacus training can help to cultivate and to improve their intelligence. They wrote a thesis to the Journal of British Medical Science. Dr. Sinagwakane, professor of Japan Medical University, Mr. Kawano and Mr. Osisewbi also did some research and made a report to the 68th Conference of Japan Physiology Association in Kyoto.

A Dissertation Submitted for the Dr. Degree in Abacus Study
Associate professor of University of Chicago, Dr.James Stigler, wrote his Doctoral Dissertation about abacus study. He stayed in Taiwan for two years and did research to the students. He gave a test to the students before and after taking abacus lessons. Then he wrote a doctoral dissertation and received his PhD. His major is in Psychology in Human Behavior. Later, he wrote many theses, including the "Mental Abacus - The Effect of Abacus Training on Chinese Children's Mental Calculation".

Japanese~ American Co-operative Study on the Effects on Abacus Education
Dr. Flanagan in University of Maryland and Dr.Biga, a professor in Liuqiu University cooperated to study the achievements of Japanese and American students in mathematics. The result shows that Japanese children's arithmetic scores are higher than American students are; above all, abacus learners are top in every grade.

A Guide to the Experiment on the Effects of Abacus Education
American Education wanted to know why American tend to reject arithmetic. They introduced abacus to the classrooms. Children like to work with the abacus in a concrete way instead of an abstract way. They gradually changed to be fond of studying arithmetic.

Concentration Effects
Decker Avenue School in California did a research for CONCENTRATION after introducing abacus to the children. The study indicated a very good result.

The Particular Teaching Tool
Children like to work with something concretely instead of seating and listening to lectures. Abacus method is good for active attitude toward learning, as pictured in Haslett's and piaget's theories.
Author: Mr. Kouzi Suzuki,
President of English Mental Arithmetic Education Association of Japan

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Old 11-05-2010, 04:16 PM   #2
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Neuroscientist Barry Horwitz of the National Institute on Deafness
The native language you speak may determine how your brain solves mathematical puzzles, according to a new study. Brain scans have revealed that Chinese speakers rely more on visual regions than English speakers when comparing numbers and doing sums.
Our mother tongue may influence the way problem-solving circuits in our brains develop, suggest the researchers. But they add that different teaching methods across cultures, or genes, may also have primed the brains of Chinese and English speakers to solve equations differently.
The findings may help educators to identify the best way to teach young students maths. Leaders of North American engineering schools and technology companies worry that youngsters in the region lag behind those in China and Japan in terms of computational skills.
Research published in 2001 has fuelled their concerns: a study comparing Canadian and Chinese students found that the latter were better at complex maths (Journal of Experimental Psychology, vol 130, p 299). But experts have wanted basic information about how brain function differs between the groups.
College seniors
In the latest study, researchers led by Yiyuan Tang at Dalian University of Technology, China, recruited 12 local college seniors in the northeastern city of Dalian, where Mandarin is spoken.
The team also recruited 12 native English speakers from the US, Australia, Canada and England to teach in Dalian. All participants were in their twenties, and both groups had equal numbers of women and men.
The volunteers lay in a magnetic resonance imaging (MRI) brain scanner while solving maths puzzles. They had to push a button, for example to indicate whether a third digit was equal to the sum of the first two numbers presented to them.
All tests were conducted using Arabic numerals, used by English-speaking cultures and taught to Chinese students at an early age.
The brain scans showed similar activity in the parietal cortex of both groups’ brains, a region thought to give a sense of quantity.
Additional areas
“But native English speakers rely more on additional brain regions involved in the meaning of words, whereas native Chinese speakers rely more on additional brain regions involved in the visual appearance and physical manipulation of numbers,” says Eric Reiman of the Banner Good Samaritan Medical Center in Phoenix, Arizona, US, one of the team.
Specifically, Chinese speakers had more activity in the visual and spatial brain centre called the visuo-premotor association network. Native English speakers showed more activity in the language network known as perisylvian cortices in the left half of the brain.

Reiman and his colleagues suggest that the Chinese language’s simple way of describing numbers may make native speakers less reliant on language processing when doing maths. For example, “eleven” is “ten one” in Chinese “twenty-one” is “two ten one”
They also note that the use of the abacus in many Asian schools may encourage the brains of students in this region to think spatially and visually about numbers.

“The results do suggest that learning to read in a particular way - or more generally, the cultural differences associated with different language groups - may have an impact in other cognitive domains, in this case arithmetic processing,” comments neuroscientist Barry Horwitz of the National Institute on Deafness and other Communication Disorders in Bethesda, Maryland, US.
Reaction times
Reiman and his colleagues found no significant difference in the reaction time and accuracy of the Chinese and English-speaking volunteers.
Still, experts believe the study opens doors to explore the causes and consequences of brain differences in mathematical processing across cultures.

I think this study adds to a number of others that suggest that brain imaging may start to have an impact on education,” Horwitz says. “By determining that not everybody learns in the same way, it may allow us to develop educational methods that work more effectively.”Some experts say that the findings of the new study may convince US educators to try introducing the abacus into more maths lessons.Journal reference: Proceedings of the National Academy of Sciences (DOI: 10.1073/pnas.0604416103)

Hand waving boosts mathematics learning
11:48 18 February 2006
NewScientist.com news service
Roxanne Khamsi, St Louis

Gestures that complement rather than simply illustrate verbal instructions can boost children's ability to complete problems in mathematics, researchers report.
"The teachers are giving the kids two different approaches to the problem - one by hand and one by mouth - and somehow they seem to complement one another," says Susan Goldin-Meadow of the University of Chicago, US. She adds that early findings also show that students who copy the gestures of their teachers are more likely to learn.
Goldin-Meadow and her colleagues gave 160 children between the ages of 8 and 10 a set of mathematical problems to solve. The students were randomly assigned to receive either verbal instructions alone or also with gestures. Those in the latter group either received gestures that copied or complemented the spoken guidance.
As part of the experiment students had to complete the equation “7+6+5=?+5”. Teachers told the youngsters that they had to make one side of the equation match the other side.
The gestures simply duplicating these directions involved the instructors pointing to the left-hand and then the right-hand sides of the equation. When using complementary gestures, however, the teachers pointed to each of the numbers on the left and then signalled the subtraction of the five on the right side by scooping their hand away from the number.
Sign of success
Children who saw the complementary gestures did best, solving three of the four addition problems correctly, on average. By comparison, those children who witnessed simple illustrative gestures typically solved fewer than two of the problems correctly. And students who received only verbal instructions solved only one of the four problems correctly, on average.Hannes Vilhjalmsson of the University of California, Los Angeles, US, who studies the use of gestures, says that the results are important as one would not expect complementary hand signals to be more helpful than reinforcing signals. "It's counter-intuitive," he says.
The work presented by Goldin-Meadow at the 2006 American Association for the Advancement of Science annual meeting in St Louis, Missouri, on Friday also suggests that children also learn better when they use gestures as well. "When we get them to gesture more it turns out that they learn more, so gesture, in general, is good for learning," she says.

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Old 11-05-2010, 04:40 PM   #3
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Conclusions about brain-based factors in dyscalculia.

We know that children with developmental arithmetic disorders can show selective deficits in specific arithmetical components, which can parallel the deficits found in brain damaged patients. Some children experience arithmetic disorders as part of genetic syndromes, or early acquired brain damage. Though such causes are rare, there appears to be a genetic component to many developmental arithmetical problems that are not caused by specific syndromes. All of this would suggest a physiological, probably brain based contribution to developmental arithmetical deficits (this does not mean that they are caused exclusively by physiological factors). There is evidence for an association in some groups of children between arithmetical deficits and reduced size of the left parietal lobe.

Arithmetical difficulties in relation to verbal and spatial ability

Reasoning, including arithmetical reasoning, can be carried out in many ways. Two broad categories that are often discussed with regard to individual differences are verbal and spatial reasoning. Information can be represented, manipulated and analysed in words; it can also be represented, manipulated and analysed in terms of visual-spatial imagery.

A number of researchers have investigated the issue of whether arithmetical skills are particularly associated with verbal or spatial reasoning and/or with discrepancies between the two. The factor analytic studies used to construct the IQ scales have consistently placed the Arithmetic subtest (one which emphasizes word problem solving) within the Verbal scale. However it has sometimes been suggested that spatial difficulties are particularly associated with difficulties in arithmetical reasoning. Rourke (1993; Strang and Rourke, 1983) proposed that verbal weaknesses lead to memory difficulties and that nonverbal weaknesses lead to logical difficulties.

He proposed two basically different groups of children with arithmetical learning disabilities. Children in the first group have difficulties in retrieval of number facts and in working memory, but have a reasonably good understanding of number concepts. They have higher nonverbal than verbal IQs and often have difficulties with reading as well as mathematics. Children in the second group do not have memory problems but do have conceptual problems; they have higher verbal than nonverbal IQs; are less likely to have reading or language difficulties, but more likely to have spatial and social difficulties associated with right hemisphere deficits.

A few studies by Rourke and others (e.g. Robinson, Menchetti and Torgesen, 2002) have supported the view that children with both reading and mathematical deficits tend to have more memory difficulties but fewer conceptual difficulties than those with just mathematical deficits.

However, there has been no consistent support for the view that 'left hemisphere'-type verbal deficits are associated with procedural and factual memory difficulties in arithmetic, while 'right-hemisphere'-type nonverbal deficits are associated with conceptual difficulties in arithmetic. Shalev, Manor, Amir, Weirtman and Gross-Tsur (1997) found no differences in the types of mathematical difficulty demonstrated by dyscalculic children with higher verbal versus higher non-verbal IQ.

Jordan and Hanich (2000) studied 76 American second-grade children were studied. They were divided into four achievement groups: 20 children with normal achievement in reading and mathematics; 10 children with difficulties in both reading and mathematics (MD-RD), 36 children with difficulties in reading only (RD) and 10 children with difficulties in mathematics only (MD). They were given tests of four areas of mathematical thinking: number facts, story problems, place value and written calculation. Children with MD/RD performed worse than NA children on all aspects of mathematics; those with MD performed worse than NA children only on story problems.

Hanich, Jordan, Kaplan and Dick (2001) similarly divided 210 second-graders were divided into four achievement groups: children with normal achievement in reading and mathematics; children with difficulties in both reading and mathematics (MD-RD), children with difficulties in reading only (RD) and those with difficulties in mathematics only (MD). Both MD groups performed worse than the other groups in most areas of arithmetic. The MD-only group outperformed the MD-RD group in both exact mental calculation and problem solving. The two MD groups performed similarly on written calculation, place value understanding, and approximate arithmetic.

Geary, Hoard and Hamson (1999) studied 90 first-grade children in the average IQ range. They included 35 children with normal achievement in reading and mathematics (N); 15 children with mathematical difficulties (MD; as shown by scores below the 30th percentile on the Mathematical Reasoning subtest of the Wechsler Individual Achievement Test); 15 children with reading difficulties (RD; as shown by scores below the 30th percentile on the Word Attack subtest of the Woodcock Johnson Psycho-Educational Battery; and 25 children with both mathematical and reading difficulties (MD/RD). Both MD groups showed problems in fact retrieval and in using counting strategies correctly in arithmetic. Children who had difficulties with both mathematics and reading tended to show problems in understanding counting principles and detecting counting errors; those with only MD or RD did not. However, about half of the MD children made double-counting errors. The MD/RD children, and those MD children who made double-counting errors, had lower backward digit spans than the other children.

Thus, the studies by Jordan and her colleagues and by Geary et al (1999) suggest that children with combined mathematical and reading disabilities tend to perform badly on more aspects of mathematics than children who only have mathematical difficulties; but do not support the type of dichotomy suggested by Rourke.

There is still less evidence that, within the general population, verbal and nonverbal ability are associated with consistently different forms of strengths and weaknesses within arithmetic. (This is not to say that there might not be such patterns within the broader domain of mathematics; e.g. geometry is likely to be more specifically associated with spatial ability than is arithmetic). Dowker (1995, 1998) looked at WISC IQ scores, calculation and derived fact strategy use in 213 children between the ages of 6 and 9. Both Verbal and Performance I.Q. predicted performance on tasks of both arithmetical calculation and derived fact strategy use. Verbal I.Q. was a stronger predictor than Performance I.Q. of both types of arithmetical task. Children who showed a strong discrepancy between verbal and nonverbal I.Q. in either direction tended to do well at tasks that involve the use of derived fact strategies; such discrepancies did not predict calculation performance.

Mathematical difficulties in children with language problems and dyslexia

Although there is no clear association between relative verbal versus spatial strengths and particular types of mathematical disability, there is no doubt that mathematical difficulties often co-occur with dyslexia and other forms of language difficulty.

People with dyslexia usually experience at least some difficulty in learning number facts such as multiplication tables. Miles (1993) found that 96% of a sample of 80 nine-to-twelve-year-old dyslexics had were unable to recite the 6x, 7x and 8x tables without stumbling.
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Old 11-05-2010, 04:54 PM   #4
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[Miles, Haslum and Wheeler (2001) used data from the British Births Cohort Study of 12,131 children born in England, Wales and Scotland between April 5th and 11th, 1970. The children were given a word recognition test, the Edinburgh Reading Test of reading comprehension, the British Abilities Scales spelling test, and the Similarities and Matrices 'intelligence' subtests of the British Abilties Scales. The children were categorized as normal achievers (49% of the sample; IQ scores of at least 90, and no significant mismatch between IQ, reading and spelling); low ability children (25% of the sample; IQ scores below 90); moderate underachievers (13% of the sample; reading and/or spelling score 1 to 1.5 standard deviations below the prediction); and severe underachievers (7% of the sample; reading and/or spelling score more than 1.5 standard deviations below the predictions). 6% were excluded due to insufficient data. 269 of the 907 severe underachievers were considered as probable dyslexics, on the grounds of poor performance on a digit span test, and on the Left-Right, Months Forwards and Months Reversed subtests of the Bangor Dyslexia Test. These dyslexic children performed less well on average on a calculation task, the Friendly Maths Test, than the normal achievers, and even than underachievers who did not meet full criteria for dyslexia. Items that were particularly difficult for the dyslexics were those which involved several steps (e.g. borrowing from two columns, and thus placed a heavy load on working memory; and those which involved fractions and decimals.

Yeo (2001) is a teacher at Emerson House, a school for dyslexic and dyspraxic primary school children, and has written extensively about the mathematical difficulties of some dyslexic children. She reports that while many dyslexic children have difficulties only with those aspects of arithmetic that involve verbal memory, some dyslexic children have more fundamental difficulties with 'number sense'. They comprehend numbers solely in terms of quantities to be counted and do not understand them in more abstract ways, or perceive the relationships between different numbers. Yeo suggests that the counting sequence presents so much difficulty for this group that it absorbs their attention and prevents them from considering other aspects of number. This sort of difficulty occurs in some children who are not dyslexic (see section above on “Common types of arithmetical difficulty); and at present the extent to which it characterizes dyslexics more than others is not clear.

Children with specific language impairment usually have some weaknesses in arithmetic, but once again some components tend to be affected much more than others. Fazio (1994) compared 20 5-year-olds with diagnosed specific language impairments with 20 age-matched controls and 20 language-matched younger children.

The language-impaired children resembled the younger children in the range and accuracy of their counting, but the age-matched controls in their understanding of counting-related concepts, such as the fact that the last item in a count sequence indicates the number of items in the set. Two years later, Fazio (1996) followed up 16 of the language-impaired children, 15 of the age-matched controls and 16 of the language-matched controls. The
language-impaired children were still poor at verbal counting, but resembled their age-matched controls in counting objects, and in reading numerals. They were worse at calculation than the age-matched controls, but worse than the language-matched controls.

Grauberg, E. (1998) has concluded from the research, and from her own experience in teaching pupils with language disorders, that they tend to have difficulties in particular with:

(1) Symbolic understanding. This includes difficulty in understanding how one item can ‘stand for’ another item or items, and effects can range from difficulties in understanding how a numeral can represent a quantity to difficulties in understanding how a coin of one denomination may be equivalent to a set of coins of a smaller denomination. Typically developing children under the age of 4 may have problems in distinguishing the cardinal use of numbers to represent quantities from their use as labels (“I am four”; “I live at number 63”). For children with language difficulties, such problems can persist for far longer. Place value – the use of the position of a digit to represent its value – can present problems for any child, but such problems are likely to be far greater for those with language disorders.

(2) Organization. Children with language disorders often have difficulties with organizing items in space or time, which may, for example, affect their ability to arrange quantities in order; to organize digits spatially on a page; and to ‘talk through’ a problem, especially a word problem.

(3) Memory. Poor short-term and long-term verbal memory are frequent characteristics of individuals with language disorders (see studies quoted above) and will affect learning to count, remembering number facts, and keeping track of one step in an arithmetic problem while carrying out subsequent steps.

In addition, language difficulties will directly affect the child’s ability to benefit from oral or written instruction, and to understand the language of mathematics.

Arithmetic in people with general learning difficulties

There are certain forms of brain damage and of genetic disorder (e.g. Williams syndrome) which not only lead to general intellectual impairment, but to disproportionate difficulties in arithmetic. There are also some people with general intellectual impairments who nevertheless perform well at arithmetic: extreme examples are savant calculators (Heavey, 2003). In general, however, even people with severe intellectual impairments tend to show similar arithmetical performance and strategies to typically developing individuals of the same mental age (Baroody, 1988; Fletcher, Huffman, Bray and Grupe, 1998).
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Old 11-05-2010, 05:02 PM   #5
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General learning difficulties are not the major topic of this review. It is, however, important to consider the topic briefly, as (a) dyscalculia is often defined in terms of the absence of general learning difficulties; and (b) there may be commonalities in some of the arithmetical deficits shown but the two groups, which could make some pooling of research desirable.

Hoard, Geary and Hamson (1999) compared 19 American first grade children with low IQs (mean 78; s.d. 5.6) with 43 children with average or above-average IQ (mean 108; s.d. 11.3). The low-IQ group showed lower backward digit span and slower articulation rates for familiar words, which may suggest working memory deficits. They were less good at number naming and number writing and magnitude comparisons. They performed worse than their peers at detecting counting errors, especially when set sizes increased beyond 5.

They made more errors in simple addition, but used a similar range of strategies: an interesting point when one remembers that they were being compared with children of similar chronological age.

We have established that arithmetical difficulties often but not always occur in people who either have generally low IQs or relatively specific reading difficulties (dyslexia). We have also established that many people have arithmetical difficulties that are not associated with either low IQ or dyslexia. Does the nature or severity of the arithmetical difficulties actually differ according to their level of specificity?

One study suggested that the level of specificity may not in fact be important in predicting the nature of the arithmetical difficulties. Gonzalez and Espinel (1999) found that children whose arithmetical achievement was much worse than would be predicted from their IQ did not differ much in their arithmetic performance from those whose poor arithmetic performance was consistent with below-average IQs. The two performed similarly on addition and subtraction word problem solving tasks and on some working memory tasks.

Thus, it appears that distinguishing specific arithmetical difficulties from difficulties associated with low IQ is important from the point of view of understanding a child’s general educational needs. Some of the arithmetical interventions needed may in fact be similar in those with specific and non-specific arithmetical deficits; though specifically dyscalculic children’s good general reasoning abilities may be used in helping children to develop compensatory arithmetical strategies.

Interventions for children with dyscalculia

Interventions are important for children with dyscalculia. These may be useful at any stage where the problem is discovered. Nonetheless, they should ideally take place as early as possible.

This is not because of any 'critical period' or rigid timescale for learning. There is no evidence for such a critical period in mathematics learning: for example, age of starting formal education has little impact on the final outcome (TIMSS, 1996).

Nonetheless, there is one important potential constraint on the timescale for learning arithmetic and other aspects of mathematics (apart, of course, from the practical constraints imposed by school curricula and the timing of public examinations). Many people develop anxiety about mathematics, which can be a distressing problem in itself, and also inhibits further progress in the subject (Fennema, 1989; Hembree, 1990; Ashcraft, Kirk and Hopko, 1998). This is rare in young children (Wigfield and Meece, 1988) and becomes much more common in adolescence. Intervening to improve arithmetical difficulties in young children may reduce the risk of later development of mathematics anxiety. In any case, interventions will be easier to carry out if they take place before mathematics anxiety has set in.

Crucially when planning interventions, there is by now overwhelming evidence that arithmetical ability is not unitary: it is made up of many components, ranging from knowledge of the counting sequence to estimation to solving word problems. Moreover, though the different components often correlate with one another, weaknesses in any one of them can occur relatively independently of weaknesses in the others. Several studies have suggested that it is not possible to establish a strict hierarchy whereby any one component invariably precedes another component.

The componential nature of arithmetic is important in planning and formulating interventions with children who are experiencing arithmetical difficulties. Any extra help in arithmetic is likely to give some benefit. However, interventions that focus on the particular components with which an individual child has difficulty are likely to be more effective than those which assume that all children's arithmetical difficulties are similar (Weaver, 1954; Dowker, 1998, 2003).

Taking mathematical difficulties into account within the classroom situation

There are by now several guides for teachers, influenced both by research findings and by teachers' reported experience, regarding strategies for dealing with individual differences within a class, and including children with mathematical difficulties.

Implications for general classroom practice (p.49) include such issues as 'including something to see, something to listen to, and something to do, at each new stage of mathematical development'; 'capitalizing on classroom opportunities for group discussion and discussion'; 'allowing plenty of classroom opportunities for discussion'; 'rehearsing, as appropriate, earlier stages prior to the introduction of new stages and challenges'; etc.

So far, books about teaching children with mathematical difficulties have tended to focus on difficulties that are associated with dyslexia (e.g. Miles and Miles, 1992; Chinn and Ashcroft, 1998; Kay and Yeo, 2003; Yeo, 2003). Thus, they tend to focus on methods of compensating for and overcoming difficulties associated with weaknesses in verbal memory. For example, Kay and Yeo (2003) suggest that, rather than attempting to learn multiplication tables verbally by rote, dyslexic pupils might use rehearsal cards, which include mathematical facts (5 x 4 =20) or definitions ('Multiply' means 'groups of' or 'times' or 'x'.), Each individual child may be given a small set of cards to practice each day under adult supervision.
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Old 11-05-2010, 05:17 PM   #6
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El-Naggar (1996) focusses on mathematical learning difficulties more generally, and discusses both ways in which individualized programmes can be used in a classroom setting, and ways in which general classroom practice may assist those with special difficulties. She points out that individualized programmes do not necessarily require one-to-one teaching. They do involve assessment of the child’s individual needs, and providing for these needs, for example by (i) small-group activities including several children with difficulties; (ii) including activities geared at the whole class revising and consolidating earlier learning in areas which the child with difficulties needs to master; (iii) providing classroom activities which can be solved at several levels; e.g. with concrete materials by children with difficulties and in more abstract form by children without such difficulties.

Intervention programs

The review will now discuss some specific intervention programs that have been used. It will focus in particular on targeted interventions for individuals and small groups of children with diagnosed arithmetical difficulties. Most such programs do not make a sharp distinction between dyscalculia and other forms and degrees of arithmetical difficulty. It is likely that such programs would prove of use for pupils with dyscalculia; but it is important that their relative success with different groups should be carefully evaluated.

Intervention programs at preschool level

It may be noted that there are now a number of preschool intervention programs for children at risk, usually children living in poverty. These appear to be commonest in the United States and include the mathematical components of the Head Start program (Arnold, Fisher, Doctoroff and Dobbs, 2002); the Berkeley Maths Readiness Project (Starkey and Klein, 2000); the Rightstart program of Griffin, Case and Siegler (1994); and the Big Math for Little Kids program of Ginsburg, Balfanz and Greenes (1999). Similar programs in Britain include the mathematical components of the PEEP program (Roberts, 2001), and the Family Numeracy program recently instituted by the British Government.

These projects involve introducing mathematical activities and games into the preschool curriculum, and in some cases also training the parents to use educational materials at home. Such programs have had promising results so far. They will not received further discussion in this review as they are not specifically targeted at children with mathematical difficulties. However, if we can develop appropriate methods of diagnosing and predicting mathematical difficulties at an early stage, such techniques could well be adapted to ameliorate such difficulties early on, with a view to reducing subsequent problems.

Van Luit and Schopman (2000) carried out one such study in the Netherlands. They examined the effects of early mathematics intervention with young children attending kindergartens for children with special educational needs. The participants were 124 children between the ages of 5 and 7. They did not have sensory or motor impairments, or severe general learning disabilities. Most had language deficits and/ or behavioural problems. All had scored in the lowest 25% for their age group on the Utrecht Test for Number Sense, a test of early counting skills and number concepts. 62 underwent intervention, and the other 62 served as a control group, who underwent the standard preschool curriculum. The intervention program was the Early Numeracy Program, which designed for children with special needs, and emphasizes learning to count. The program involved the numbers 1 to 15, which were represented in various ways, progressing from the concrete (sets of objects) through the semi-concrete (tallies) to the abstract (numerals) sets of objects, and tally marks. Patterns of 5 were particularly emphasized, and were represented by 5 tally marks within and ellipse. The number activities were embedded in games involving families, celebrations and shopping. The children had two half-hour sessions per week in groups of three for six months. At the end, the intervention group performed much better than the control group on activities that had formed part of the intervention program, but unfortunately did not transfer their superior knowledge to other similar but not identical numeracy tasks.

Individualized remediation with school-aged children with arithmetical difficulties

We now turn from group-based techniques of helping children with arithmetical difficulties to more individualized component-based techniques, that take into account individual children's strengths and weaknesses in specific components of arithmetic. Some of these projects are totally individual; some include at least some small-group work.

Assessments for targeted intervention

Effective interventions imply some form of assessment, whether formal or informal, to (a) indicate the strengths, weaknesses and educational needs of an individual or group; and (b) to evaluate the effectiveness of the intervention in improving performance.

There are a variety of standardized tests used for assessing children's arithmetic. Many test batteries for measuring abilities (e.g. the British Abilities Scales and their American counterpart, the Differentiated Aptitude Tests) include tests both of calculation efficiency and of mathematical reasoning, the latter usually taking the form either of number pattern recognition or word problem solving. IQ scales, such as the Wechsler Intelligence Scale for Children and the Weschler Adult Intelligence Scale, include arithmetic subtests which tend to emphasize word problem solving. Some tests, used in school contexts (e.g. the NFER Mathematics tests, and the SATS), place greater emphasis on whether children have mastered particular aspects of the arithmetic curriculum. Others are devised by researchers for the specific purpose of assessing particular mathematical components, which are to be dealt with, or have been dealt with, in an intervention program. Some researchers over the years have argued that the exclusive use of standardized tests may result in missing crucial aspects of an individual's strategies and difficulties, and have emphasized the importance of individual interviews and case study methods (Brownell and Watson, 1936; Ginsburg, 1977).

Most assessment techniques involve testing children across the range of ability, and dyscalculia is diagnosed by a score below a certain cut-off point in mathematical tests, without similarly low scores in other tests.

Butterworth (2002; Butterworth, in press) has devised a computerized screening test of basic numerical skills, which is more specifically directed at incorporating the recognition of small numerosities; estimation of somewhat larger numerosities; and comparisons of number size. These are intended to identify severe arithmetical difficulties (dyscalculia) rather than to assess individual differences in the general population.

Some of the history of individualized remedial work

It is striking how many of the most modern practices have surprisingly early origins. Some forms of individualized, component-based techniques of assessing and remediating mathematical difficulties have been in existence at least since the 1920s (Buswell and John, 1927; Brownell, 1929; Greene and Buswell, 1930; Williams and Whitaker, 1937; Tilton, 1947). On the other hand, they have never been used very extensively; and there are many books, both old and new, about mathematical development and mathematics education, which do not even refer to such techniques, or to the theories behind them.

Weaver (1954) was a strong advocate of differentiated instruction and remediation in arithmetic. He put forward several important points that have since been strongly supported by the evidence, centrally that "arithmetic competence is not a unitary thing but a composite of several types of quantitative ability: e.g. computational ability, problem-solving ability, etc."; that "(t)hese abilities overlap to varying degrees, but most are sufficiently independent to warrant separate evaluations"; and that "children exhibit considerable variation in their profiles or patterns of ability in the various patterns of arithmetic instruction" (pp. 300-301). He argued (pp. 302-303) that any "effective program of differentiated instruction in arithmetic must include provision for comprehensive evaluation, periodic diagnosis, and appropriate remedial work" and that "(e)xcept for extreme cases of disability, which demand the aid of clinicians and special services, remedial teaching is basically good teaching, differentiated to meet specific instructional needs".

For a long time, some researchers and educators have emphasized the importance of investigating the strategies that individual children use in arithmetic: especially those faulty arithmetical procedures that lead to errors (Buswell and John, 1926; Brownell, 1929; Van Lehn 1990). Thus, some children might add without carrying (e.g. 23 + 17 = 310); others might add all the digits without any reference to whether they are tens or units (e.g. 23 + 17 = 13); others, when adding a single-digit number to a two-digit number, might add it to both the tens and the units (e.g. 34 + 5 = 89). Much of the work over the years has looked at the faulty arithmetical procedures that children often demonstrate: e.g. when subtracting


a common faulty procedure is to always subtract the smaller number from the larger, in this case obtaining the answer 36.
Another faulty procedure is to omit borrowing and to write 0 when a larger digit seems to be subtracted from a smaller:

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Old 11-05-2010, 05:37 PM   #7
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The children in the project, together with some of their classmates and children from other schools, are given three
standardized arithmetic tests: the British Abilities Scales Basic Number Skills subtest (1995 revision), the WOND Numerical Operations test, and the WISC Arithmetic subtest. The first two place greatest emphasis on computation abilities and the latter on arithmetical reasoning. The children are retested at intervals of approximately six months.

The children in the intervention group have so far shown very significant improvements. (Average standard scores are 100 for the BAS Basic Number Skills subtest and the WOND Numerical Operations subtest, and 10 for the WISC Arithmetic subtest.) The median standard scores on the BAS Basic Number Skills subtest were 96 initially and 100 after approximately six months. The median standard scores on the WOND Numerical Operations test were 91 initially and 94 after six months. The median standard scores on the WISC Arithmetic subtest were 7 initially, and 8 after six months (the means were 6.8 initially and 8.45 after six months). Wilcoxon tests showed that all these improvements were significant at the 0.01 level.

One hundred and one of the 146 children have been retested over periods of at least a year, and have been maintaining their improvement.

Governments and targeted intervention programs

A positive and interesting development is that some Governments are developing and using targeted and individualized interventions for children with mathematical difficulties. These include Britain (DfEE, 2003) and at least some Australian states (Kraayenoord and Elkins, 2004).

Computer programs for individual instruction

With the increasing development and availability of computer technology, a number of computer programs have been developed individualized instruction and remedial work as other individualized self-teaching systems. In addition, they have the important advantage that computers are motivating to many children; and that, with increasing availability of home computers and computer games, they may be used outside of as well as within a school context.

Computer programs in the past tended to take a simplistic approach to children's errors and to reward correct answers, and reject incorrect answers, without scope for analyzing how the errors occurred (Hativa, 1988). The more sophisticated forms of programming that are available today make it much more possible to diagnose and interpret misconceptions; though, as with any test, they may not pick up a particular individual's interpretations and misinterpretations, especially if these are somewhat untypical of the population as a whole.

Most studies of computer-based intervention with children with mathematical disabilities are as yet relatively small-scale, involving small samples. Recent results have been quite promising (Errera et al, 2001; Earl, 2003; Pennant, 2003).

It should be noted, however, that Kroesbergen and Van Luit's (2003) meta-analysis of mathematical training and intervention studies indicated that computer-based interventions tend to result in less progress than interventions carried out by teachers. These results may be based in part on sampling differences, and certainly do not mean that computer-based interventions are worthless; however, they should not be seen as a replacement for interventions by human beings. Computer-based interventions, in any case, take many forms and some will be more effective than others. In any case, they have the potential to serve very useful purposes in increasing motivation and reducing the impact of emotional, communication, or motor difficulties.

Implications for further research

The intervention studies so far have involved a rather heterogeneous group of children with mathematical difficulties, who have been differentiated according to their level of functioning in various components of mathematics, but not on the basis of whether they have a specific or non-specific mathematical deficit. It would be desirable to compare the effectiveness of such interventions with (a) children with dyscalculia; (b) children with milder, possibly more environmentally caused, specific arithmetical deficits; (c) children with combined dyscalculia and dyslexia; and (d) children with arithmetical deficits as part of more general learning difficulties.

Several researchers (Mazzocco and Myers, 2003; Desoete, Roeyers and DeClercq, 2004) have emphasized the fact that dyscalculia can vary considerably, both in incidence and prognosis, according to the exact criteria used: (e.g. absolute level of deficit; deviation from IQ level; and/ or persistence over time in the early years). It is important to establish the criteria being used in a study, and also to compare results when different criteria are used.

Further research is also of course necessary to show whether and to what extent the individualized interventions described here are more effective in improving children's arithmetic than other interventions which provide children with individual attention: e.g. interventions in literacy, or interventions in arithmetic which are conducted on a one-to-one basis but not targeted toward individual strengths and weaknesses.

Other types of intervention are also worthy of further study: in particular, computer-based interventions with dyscalculic children are still virtually in their infancy, and are worthy of further development.

It is important to compare the different programs using similar forms of assessment. At present, as pointed out by Kroesbergen and Van Luit (2003) and by Rohrbeck et al (2003), it is difficult to compare programs, because most researchers and project managers have worked in relative isolation, unaware of each other’s programs. Most programs have involved different methods of sampling and different forms of assessment, rendering it difficult or impossible to make valid comparisons.

Another goal of research should be to investigate the role of targeted interventions for adults with mathematical difficulties. Most intervention programs have been with children or adolescents. Since numeracy difficulties have lifelong implications, it is important that more work be carried out on diagnosis and intervention for such difficulties in adults. Numeracy is increasingly included in 'basic skills' programs for adults; though most such programs do not differentiate between adults who have not learned such skills due to lack of educational opportunity, and those who have dyscalculia.

Given the importance of preschool interventions with at-risk children, it would be desirable to have more investigations of methods of assessing preschool children’s early mathematical abilities; of predicting different forms of mathematical difficulty; and of targeting early interventions to have maximum impact in preventing, or at least reducing the subsequent impact, of such difficulties.

Greater communication and collaboration between scientists, teachers and policy-makers is vital. This was indeed pointed out by Piaget (1971), but has only rarely been put into practice.
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Old 12-05-2010, 04:01 PM   #8
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Abacus Experts Neural Correlates Underlying Mental Calculation In: A Functional Magnetic Resonance Imaging Study

Takashi Hanakawa, Manabu Honda, Tomohisa Okada, Hidenao Fukuyama and Hiroshi Shibasaki Human Brain Research Center, Kyoto University Graduate School of Medicine, Kyoto, Japan Human Motor Control Section, NINDS, NIH, Bethesda, MD 20892, USANational Institute for Physiological Sciences, Okazaki, Japan PRESTO, Japan Science and Technology Corp., Kawaguchi, JapanDepartment of Nuclear Medicine, Kyoto University Graduate School of Medicine, Kyoto, Japan

Experts of abacus operation demonstrate extraordinary ability in mental calculation. There is psychological evidence that abacus experts utilize a mental image of an abacus to remember and manipulate large numbers in solving problems; however, the neural correlates underlying this expertise are unknown. Using functional magnetic resonance imaging, we compared the neural correlates associated with three mental-operation tasks (numeral, spatial, verbal) among six experts in abacus operations and eight nonexperts. In general, there was more involvement of neural correlates for visuospatial processing (e.g., right premotor and parietal areas) for abacus experts during the numeral mental-operation task. Activity of these areas and the fusiform cortex was correlated with the size of numerals used in the numeral mental-operation task. Particularly, the posterior superior parietal cortex revealed significantly enhanced activity for experts compared with controls during the numeral mental-operation task.

Comparison with the other mental-operation tasks indicated that activity in the posterior superior parietal cortex was relatively specific to computation in 2-dimensional space. In conclusion, mental calculation of abacus experts is likely associated with enhanced involvement of the neural resources for visuospatial information processing in 2-dimensional space.

Fig. 1. Abacus operation and behavioral tasks. (A) The basic operation of a Japanese abacus or “soroban.” It has columns of beads, and each of them has a place value, corresponding to the ones, tens, hundreds, and so on (i.e., base-10 system of numeration). Once a specific column of beads is arbitrarily defined as the “ones,” the other columns are valued relative to it. Each column of beads has an upper and a lower section. The bead in the upper section is equal to 5 times the unit value of the columns when it is pushed down toward the horizontal dividing bar, and each of the 4 lower beads is equal to 1 unit value of the columns when pushed up. For example, for the addition of 9 and 7, 4 lower beads up and 1 upper bead down in a column represent the number 9. To execute the addition requiring a carrying, one first subtracts the complement of the addend to 10 (3 here) and then adds 1 to the tens column. One can accomplish this calculation using a single finger movement, twisting the thumb (T) and the index finger (I) to push up 1 bead in the tens column (white arrow) and to push down 3 beads in the ones column (black arrow), respectively. (B) In the numeral mental-operation task, subjects were asked to add a series of numbers presented visually. (C) In the spatial mental-operation task, subjects were required to mentally move the marker location according to an arrow or a pair of arrows. For example, one should move the marker 1 square to the right, as indicated by the single arrow pointing to the right, and then down 2 squares, as indicated by the double arrows pointing down, and so forth. (D) In the verbal mental-operation task, subjects first remembered the day of the week specified by a kanji character and advanced the date as instructed by the numbers. For example, one first advanced the day from Wednesday to Friday by the number 2, from Friday to Monday by the number 3, and so forth.

Fig. 2. General pattern of activity during each mental-operation task. (A) Activity during the three mental-operation tasks relative to a visual fixation task for abacus experts and nonexperts (within-group conjunction analysis, P < 0.05 corrected for multiple comparisons), rendered onto a standard brain. Significant activity only for abacus experts is shown in red, activity only for nonexperts in green, and the spatial overlap of activity in yellow. (B) Laterality of activity for the superior precentral sulcus (SPcS) and intraparietal sulcus (IPS), where the three mental-operation tasks commonly induced activity. A liberal threshold of P < 0.001 (uncorrected) was used only to locate these activities in each individual; all subjects showed suprathreshold activity in all four regions at this threshold. Estimated mean signal increase (SI, arbitrary unit) at the most significantly activated single voxel was computed in each area in each individual and then was used to calculate a laterality index (LI), that is SI for left minus SI for right, divided by the sum of SI for both sides. The more involvement of the right hemisphere was evident for experts compared with nonexperts only during the numeral mental-operation task (*P = 0.039 for SPcS and *P = 0.029 for IPS by U test).
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Old 15-05-2010, 07:05 PM   #9
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This is the first part of the best technical proof of abacus teaching value I have have ever found on the internet


James Stigler is associate professor of psychology at the University of Chicago. He was awarded the
Boyd R. McCandless Young Scientist Award from the American Psychological Association and was
awarded a Guggenheim Fellowship for his work in the area of culture and mathematics learning. Harold
W. Stevenson is professor of psychology and director of the University of Michigan Program in Child
Development and Social Policy. He is currently president of the international Society for the Study of
Behavioral Development and has spent the past two decades engaged in cross-cultural research.
This article is from American Educator, Spring 1991, and is based on a book by Harold W. Stevenson and
James Stigler entitled The Learning Gap (1994).
Although there is no overall difference in intelligence, the differences in mathematical achievement of
American children and their Asian counterparts are staggering.

Let us look first at the results of a study we conducted in 120 classrooms in three cities: Taipei
(Taiwan); Sendai (Japan); and the Minneapolis metropolitan area. First and fifth graders from
representative schools in these cities were given a test of mathematics that required computation and
problem solving. Among the one hundred first-graders in the three locations who received the lowest
scores, fifty-eight were American children; among the one hundred lowest-scoring fifth graders, sixty-
seven were American children. Among the top one hundred first graders in mathematics, there were only
fifteen American children. And only one American child appeared among the top one hundred fifth
graders. The highest-scoring American classroom obtained an average score lower than that of the lowest-
scoring Japanese classroom and of all but one of the twenty classrooms in Taipei. In whatever way we
looked at the data, the poor performance of American children was evident.

These data are startling, but no more so than the results of a study that involved 40 first- and 40 fifth-
grade classrooms in the metropolitan area of Chicago—a very representative sample of the city and the
suburbs of Cook County—and twenty-two classes in each of these grades in metropolitan Beijing
(China). In this study children were given a battery of mathematics tasks that included diverse problems,
such as estimating the distance between a tree and a hidden treasure on a map, deciding who won a race
on the basis of data in a graph, trying to explain subtraction to visiting Martians, or calculating the sum of
nineteen and forty-five. There was no area in which the American children were competitive with those
from China. The Chinese children’s superiority appeared in complex tasks involving the application of
knowledge as well as in the routines of computation. When fifth graders were asked, for example, how
many members of a stamp club with twenty-four members collected only foreign stamps if five-sixths of
the members did so, 59 percent of Beijing children, but only 9 percent of the Chicago children produced
the correct answer.
On a computation test only 2.2 percent of the Chinese fifth graders scored at or below
the mean for their American counterparts. All of the twenty Chicago area schools had average scores on
the fifth-grade geometry test that were below those of the Beijing schools. The results from all these tasks
paint a bleak picture of American children’s competencies in mathematics.
The poor performance of American students compels us to try to understand the reasons why. We
have written extensively elsewhere about the cultural differences in attitudes toward learning and toward
the importance of effort vs. innate ability and about the substantially greater amounts of time Japanese
and Chinese students devote to academic activities in general and to the study of math in particular.
Important as these factors are, they do not tell the whole story. For that we have to take a close look
inside the classrooms of Japan, China, and the United States to see how mathematics is actually taught in
the three cultures.
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Old 15-05-2010, 07:16 PM   #10
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If we were asked briefly to characterize classes in Japan and China, we would say that they consist of
coherent lessons that are presented in a thoughtful, relaxed. and nonauthoritarian manner. Teachers
frequently rely on students as sources of information. Lessons are oriented toward problem solving rather
than rote mastery of facts and procedures and utilize many different types of representational materials.
The role assumed by the teacher is that of knowledgeable guide, rather than that of prime dispenser of
information and arbiter of what is correct. There is frequent verbal interaction in the classroom as the
teacher attempts to stimulate students to produce, explain, and evaluate solutions to problems. These
characteristics contradict stereotypes held by most Westerners about Asian teaching practices. Lessons
are not rote; they are not filled with drill. Teachers do not spend large amounts of time lecturing but
attempt to lead the children in productive interactions and discussions. And the children are not the
passive automata depicted in Western descriptions but active participants in the learning process.
We begin by discussing what we mean by the coherence of a lesson. One way to think of a lesson is
by using the analogy of a story. A good story is highly organized; it has a beginning, a middle, and an
end; and it follows a protagonist who meets challenges and resolves problems that arise along the way.
Above all, a good story engages the readers’ interest in a series of interconnected events, which are best
understood in the context of the events that precede and follow it.
Such a concept of a lesson guides the organization of instruction in Asia. The curricula are defined in
terms of coherent lessons, each carefully designed to fill a forty-to fifty-minute class period with
sustained attention to the development of some concept or skill. Like a good story the lesson has an
introduction, a conclusion, and a consistent theme.
We can illustrate what we are talking about with this account of a fifth-grade Japanese mathematics
The teacher walks in carrying a large paper bag full of clinking glass. Entering the classroom with a
large paper bag is highly unusual, and by the time she has placed the bag on her desk the students are
regarding her with rapt attention. What’s in the bag? She begins to pull items out of the bag, placing them,
one-by-one, on her desk. She removes a pitcher and a vase. A beer bottle evokes laughter and surprise.
She soon has six containers lined up on her desk. The children continue to watch intently, glancing back
and forth at each other as they seek to understand the purpose of this display.
The teacher looking thoughtfully at the containers, poses a question: “I wonder which one would hold
the most water?” Hands go up, and the teacher calls on different students to give their guesses: “the
pitcher,” “the beer bottle,” “the teapot.” The teacher stands aside and ponders: “Some of you said one
thing, others said something different. You don’t agree with each other. There must be some way we can
find out who is correct. How can we know who is correct?” Interest is high, and the discussion continues.
The students soon agree that to find out how much each container holds they will need to fill the
containers with something. How about water? The teacher finds some buckets and sends several children
out to fill them with water. When they return, the teacher says: “Now what do we do?” Again there is a
discussion, and after several minutes the children decide that they will need to use a smaller container to
measure how much water fits into each of the larger containers. They decide on a drinking cup, and one
of the students warns that they all have to fill each cup to the same level—otherwise the measure won’t be
the same for all of the groups.
At this point the teacher divides the class into their groups and gives each group one of the containers
and a drinking cup. Each group fills its container, counts how many cups of water it holds, and writes the
result in a notebook When all of the groups have completed the task the teacher calls on the leader of each
group to report on the group’s findings and notes the results on the blackboard. She has written the
names of the containers in a column on the left and a scale from 1 to 6 along the bottom. Pitcher, 4.5
cups; vase, 3 cups; beer bottle, 15 cups; and so on. As each group makes its report, the teacher draws a
bar representing the amount, in cups, the container holds.
Finally, the teacher returns to the question she posed at the beginning of the lesson: Which container
holds the most water? She reviews how they were able to solve the problem and points out that the
_________________________ _______________
Page 3
answer is now contained in the bar graph on the board. She then arranges the containers on the table in
order according to how much they hold and writes a rank order on each container, from 1 to 6. She ends
the class with a brief review of what they have done. No definitions of ordinate and abscissa, no
discussion of how to make a graph preceded the example—these all became obvious in the course of the
lesson, and only at the end did the teacher mention the terms that describe the horizontal and vertical axes
of the graph they had made.
With one carefully crafted problem, this Japanese teacher has guided her students to discover—and most
likely to remember—several important concepts. As this article unfolds, we hope to demonstrate that this
example of how well-designed Asian class lessons are is not an isolated one; to the contrary, it is the
norm. And as we hope to further demonstrate, excellent class lessons do not come effortlessly or
magically Asian teachers are not born great teachers; they and the lessons they develop require careful
nurturing and constant refinement.
Elementary school mathematics is often defined in terms of mathematical symbols and their
manipulation; for example, children must learn the place-value system of numeration and the operations
for manipulating numerals to add, subtract, multiply, and divide. In addition, children must be able to
apply these symbols and operations to solving problems. In order to accomplish these goals, teachers rely
primarily on two powerful tools for representing mathematics: language and the manipulation of concrete
objects. How effectively teachers use these forms of representation plays a critical role in determining
how well children will understand mathematics.
One common function of language is in defining terms and stating rules for performing mathematical op-
erations. A second, broader function is the use of language as a means of connecting mathematical
operations to the real world of integrating what children know about mathematics. We find that American
elementary school teachers are more prone to use language to define terms and state rules than are Asian
teachers, who, in their efforts to make mathematics meaningful, use language to clarify different aspects
of mathematics and to integrate what children know about mathematics with the demands of real-world
problems. Here is an example of what we mean by a class in which the teacher defines terms and states
An American teacher announces that the lesson today concerns fractions. Fractions are defined and she
names the numerator and denominator “Wbat do we call this?” she then asks. And this?” After assuring
herself that the children understand the meaning of the terms, she spends the rest of the lesson teaching
them to apply the rules for forming fractions.
Asian teachers tend to reverse the procedure. They focus initially on interpreting and relating a real-
world problem to the quantification that is necessary for a mathematical solution and then to define terms
and state rules. In the following example, a third-grade teacher in Japan was also teaching a lesson that
introduced the notation system for fractions.
The lesson began with the teacher posing the question of how many liters of juice (colored water) were
contained in a large beaker. “More than one liter,” answered one child. “One and a half liters,” answered
another. After several children had made guesses, the teacher suggested that they pour the juice into some
one-liter beakers and see. Horizontal lines on each beaker divided it into thirds. The juice filled one
beaker and part of a second. The teacher pointed out that the water came up to the first line on the second
beaker—only one of the three parts was full. The procedure was repeated with a second set of beakers to
illustrate the concept of one-half. After stating that there had been one and one-out-of-three liters of juice
in the first big beaker and one and one-out-of-two liters in the second, the teacher wrote the fractions on
the board. He continued the lesson by asking the children how to represent two parts out of three, two
parts out of five, and so forth. Near the end of the period he mentioned the term “fraction” for the first
time and attached names to the numerator and the denominator.
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Old 16-05-2010, 11:10 AM   #11
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He ended the lesson by summarizing how fractions can be used to represent the parts of a whole.
In the second example, the concept of fractions emerged from a meaningful experience; in the first, it
was introduced initially as an abstract concept. The terms and operations in the second example flowed
naturally from the teacher’s questions and discussion; in the first, language was used primarily for
defining and summarizing rules. Mathematics ultimately requires abstract representation, but young
children understand such representation more readily if it is derived from meaningful experience than if it
results from learning definitions and rules.
Asian teachers generally are more likely than American teachers to engage their students, even very
young ones, in the discussion of mathematical concepts. The kind of verbal discussion we find in
American classrooms is more short-answer in nature, oriented, for example, toward clarifying the correct
way to implement a computational procedure.
Teachers ask questions for different reasons in the United States and in Japan. In the United States, the
purpose of a question is to get an answer. In Japan, teachers pose questions to stimulate thought. A
Japanese teacher considers a question to be a poor one if it elicits an immediate answer, for this indicates
that students were not challenged to think. One teacher we interviewed told us of discussions she had
with her fellow teachers on how to improve teaching practices. “What do you talk about?” we wondered.
“A great deal of time,” she reported. “is spent talking about questions we can pose to the class—which
wordings work best to get students involved in thinking and discussing the material. One good question
can keep a whole class going for a long time
a bad one produces little more than a simple answer.”
In one memorable example recorded by our observers, a Japanese first-grade teacher began her class
by posing the question to one of her students: “Would you explain the difference between what we
learned in yesterdays lesson and what you came across in preparing for today’s lesson?” The young
student thought for a long time, but then answered the question intelligently, a performance that
undoubtedly enhanced his understanding of both lessons.
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Old 16-05-2010, 11:20 AM   #12
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Every elementary school student in Sendai possesses a “Math Set” a box of colorful, well-designed
materials for teaching mathematical concepts: tiles, clock, ruler, checkerboard, colored triangles, beads,
and many other attractive objects.
In Taipei. every classroom is equipped with a similar but larger set of such objects. In Beijing where
their is much less money available for purchasing such materials, teachers improvise with colored paper,
wax fruit, plates, and other easily obtained objects. In all cases, these concrete objects are considered to
be critically important tools for teaching mathematics, for it is through manipulating these objects that
children can form important links between real-world problems and abstract mathematical notations.
American teachers air much less likely than Chinese or Japanese teachers to use concrete objects. At
fifth grade, for example, Sendai teachers were nearly twice as likely to use concrete objects as the
Chicago area teachers, and Taipei teachers were nearly five times as likely. There was also a subtle, but
important, difference in the way Asian and American teachers used concrete objects. Japanese teachers,
for example, use the items in the Math Set throughout the elementary school years and introduced small
tiles in a high percentage of the lessons we observed in the first grade. American teachers seek variety and
may use Popsicle sticks in one lesson, and in another marbles, Cheerios, M&Ms, checkers, poker chips,
or plastic animals. The American view is that objects should be varied in order to maintain children’s
interest. The Asian view is that using a variety of representational materials may confuse children, and
thereby make it more difficult for them to use the objects for the representation and solution of
mathematics problems. Having learned to add with tiles makes multiplication easier to understand when
the same tiles are used.
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Page 5
Through the skillful use of concrete objects, teachers are able to teach elementary school children to
understand and solve problems that are not introduced in American curricula until much later An example
occurred in a fourth-grade mathematics lesson we observed in Japan. The problem the teacher posed is a
difficult one for fourth graders, and its solution is generally not taught in the United States until much
later This is the problem:
There are a total of thirty-eight children in Akira’s class. There are six more boys than there are girls.
How many boys and how many girls are in the class?
This lesson began with a discussion of the problem and with the children proposing ways to solve it.
After the discussion, the teacher handed each child two strips of paper one six units longer than the other,
and told the class that the strips would be used to help them think about the problem. One slip represented
the number of girls in the class and the other represented the number of boys. By lining the strips next to
each other the children could see that the degree to which the longer one protruded beyond the shorter one
represented 6 boys. The procedure for solving the problem then unfolded as the teacher through skillful
questioning, led the children to the solution. The number of girls was found by taking the total of both
strips, subtracting 6 to make the strips of equal length, and then dividing by 2. The number of boys could
be found, of course, by adding 6 to the number of girls. With this concrete visual representation of the
problem and careful guidance from the teacher even fourth graders were able to understand the problem
and its solution.
A common Western stereotype is that the Asian teacher is an authoritarian purveyor of information,
one who expects students to listen and memorize correct answers or correct procedures rather than to
construct knowledge themselves. This may or may not be an accurate description of Asian high school
teachers, but, as we have seen in previous examples, it does not describe the dozens of elementary school
teachers that we have observed.
Chinese and Japanese teachers rely on students to generate ideas and evaluate the correctness of the
ideas. The possibility that they will be called upon to state their own solution as well as to evaluate what
another student has proposed keeps Asian students alert but this technique has two other important
functions. First, it engages students in the lesson, increasing their motivation by making them feel they
are participants in a group process. Second, it conveys a more realistic impression of how knowledge is
acquired. Mathematics, for example, is a body of knowledge that has evolved gradually through a process
of argument and proof. Learning to argue about mathematical ideas is fundamental to understanding
mathematics. Chinese and Japanese children begin learning these skills in the first grade; many American
elementary school students are never exposed to them.
We can illustrate the way Asian teachers use students’ ideas with the following example. A fifth-
grade teacher in Taiwan began her mathematics lesson by calling attention to a six-sided figure she had
drawn on the blackboard. She asked the students how they might go about finding the area of the shaded
region. “I don’t want you to tell me what the actual area is, just tell me the approach you would use to
solve the problem. Think of as many different ways as you can of ways you could determine the area that
I have drawn in yellow chalk.” She allowed the students several minutes to work in small groups and then
called upon a child from each group to describe the group’s solution. After each proposal, many of which
were quite complex, the teacher asked members of the other groups whether the procedure described
could yield a correct answer. After several different procedures had been suggested, the teacher moved on
to a second problem with a different embedded figure and repeated the process. Neither teacher nor
students actually carried out a solution to the problem until all of the alternative solutions had been
discussed. The lesson ended with the teacher affirming the importance of coming up with multiple
solutions. “After all,” she said, “we face many problems every day in the real world. We have to
remember that there is not only one way we can solve each problem.”
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Old 16-05-2010, 11:33 AM   #13
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American teachers are less likely to give students opportunities to respond at such length. Although a
great deal of interaction appears to occur in American classrooms—with teachers and students posing
questions and giving answers—American teachers generally pose questions that are answerable with a
yes or no or with a short phrase. They seek a correct answer and continue calling on students until one
produces it. “Since we can’t subtract 8 from 6,” says an American teacher, “we have to -. what?” Hands
go up, the teacher calls on a girl who says “Borrow.” “Correct,” the teacher replies. This kind of
interchange does not establish the student as a valid source of information, for the final arbiter of the
correctness of the student’s opinions is still the teacher. The situation is very different in Asian
classrooms, where children are likely to be asked to explain their answers and other children are then
called upon to evaluate their correctness.
Clear evidence of these differing beliefs about the roles of students and teachers appears in the
observations of how teachers evaluate students’ responses. The most frequent form of evaluation used by
American teachers was praise, a technique that was rarely used in either Taiwan or Japan. In Japan,
evaluation most frequently took the form of a discussion of children’s errors.
Praise serves to cut off discussion and to highlight the teacher’s role as the authority. It also
encourages children to be satisfied with their performance rather than informing them about where they
need improvement. Discussing errors, on the other hand, encourages argument and justification and
involves students in the exciting quest of assessing the strengths and weaknesses of the various alternative
solutions that have been proposed.
Why are American teachers often reluctant to encourage students to participate at greater length during
mathematics lessons? One possibility is that they feel insecure about the depth of their own mathematical
training. Placing more emphasis on students’ explanations necessarily requires teachers to relinquish
some control over the direction the lesson will take. This can be a frightening prospect to a teacher who is
unprepared to evaluate the validity of novel ideas that students inevitably propose.


We have been struck by the different reactions of Asian and American teachers to children’s errors.
For Americans, errors tend to be interpreted as an indication of failure in learning the lesson. For Chinese
and Japanese, they are an index of what still needs to be learned. These divergent interpretations result in
very different reactions to the display of errors—.embarrassment on the part of the American children,
calm acceptance by Asian children. They also result in differences in the manner in which teachers utilize
errors as effective means of instruction.
We visited a fifth-grade classroom in Japan the first day the teacher introduced the problem of adding
fractions with unequal denominators. The problem was a simple one: adding one-third and one-half. The
children were told to solve the problem and that the class would then review the different solutions.

After everyone appeared to have completed the task, the teacher called on one of the students to give
his answer and to explain his solution. “The answer is two-fifths,’ he stated. Pointing first to the
numerators and then to the denominators, he explained: “One plus one is two; three plus two is five. The
answer is two-fifths.” Without comment, the teacher asked another boy for his solution. “Two point one
plus three point one, when changed into a fraction adds up to two-fifths.” The children in the classroom
looked puzzled. The teacher, unperturbed. asked a third student for her solution. “The answer is five-
sixths.’ The student went on to explain how she had found the common denominator changed the
fractions so that each had this denominator and then added them.
The teacher returned to the first solution. “How many of you think this solution is correct. Most
agreed that it was not. She used the opportunity to direct the children’s attention to reasons why the
solution was incorrect. Which is larger two-fifths or one-half?” The class agreed that it was one-half. “It
is strange, isn’t it that you could add a number to one-half and get a number that is smaller than one-
_________________________ _______________
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half?” She went on to explain how the procedure the child used would result in the odd situation when,
when one-half was added to one-half, the answer yielded is one-half. In a similarly careful, interactive
manner, she discussed how the second boy had confused fractions with decimals to come up with his
surprising answer. Rather than ignoring the incorrect solutions and concentrating her attention on the
correct solution, the teacher capitalized on the errors the children made in order to dispel two common
misperceptions about fractions.
We have not observed American teachers responding to children’s errors so inventively. Perhaps
because of the strong influence of behavioristic teaching that conditions should be arranged so that the
learner avoids errors and makes only a reinforceable response, American teachers place little emphasis on
the constructive use of errors as a teaching technique. It seems likely, however that learning about what is
wrong may hasten children’s understanding of why the correct procedures are appropriate.
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Old 21-05-2010, 12:02 AM   #14
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Secondary schools can learn lessons from primary schools in Wales in how to stop pupils from falling behind in numeracy skills, says a new report.

Inspection body Estyn said secondary teachers do not place enough emphasis on mental and written calculation.

Even primary pupils overall can find it hard to recall basic maths and have problems with calculations, said Estyn.

Inspectors say pupils need to use maths in other classes to stop them falling behind.

In compiling the report, Estyn inspectors looked at four years of school inspection reports and, as well as national curriculum teacher assessments, they visited 21 schools and surveyed local authorities.

Schools should "exploit opportunities" for pupils to apply numeracy skills in non-maths subjects and "real life contexts"
Support pupils who make least progress, and making sure more-able pupils gain the higher assessment levels
More attention to improving pupils' skills in mental and written calculations
Clear policies for the appropriate use of calculators
Councils should monitor numeracy intervention programmes in schools more thoroughly
Better use of information about primary pupils' achievements in numeracy when they transfer to secondary schools
Welsh Assembly Government should continue to make funding available in grants
Ministers also need to impress on councils and schools the need to develop a strategic approach to planning numeracy intervention programmes
Source: Estyn/Improving Numeracy In Key Stage 2 and Key Stage 3
They found a widening gap between pupils' performance when aged seven to 11 - at key stage 2 - with more reaching the expected level in maths than those in secondary schools aged 11 to 14 - at key stage 3.

Inspectors want pupils to improve their numeracy skills across the subject range, and in dealing with "real life" situations.

But they found too few secondary schools used other lessons to give chances for pupils to develop numeracy skills, or build on abilities in mental and written calculations made in primary school.

Standards in mathematics are often higher than when numeracy is applied across the curriculum because pupils "don't apply numeracy skills they have learned in maths well enough outside maths lessons".

Estyn praises Pillgwenlly Primary School in Newport, which brings in numeracy skills into classes ranging from history to religious education.

In a minority of lessons, teachers of other subjects had "too low expectations" and allowed pupils to use calculators for basic calculations that pupils should do mentally.

Inspectors also found the effectiveness of "catch-up" programmes for pupils struggling with numeracy varies between local authorities and between schools across Wales.

Ysgol Bryngwyn comprehensive school in Llanelli, Carmarthenshire is praised for using teaching assistants to run "catch-up" numeracy programmes with pupils, in their last year of primary school and first year of secondary school.

Research shows that 53% of adults in Wales have numeracy skills below the level expected of an 11-year-old

Ann Keane, Estyn chief inspector
Pupils at nine primary schools are involved in classes for an hour a week, with "catch up" classes also running for up to 30 pupils continuing in their first year of secondary school.

Dr Margaret Williams, Bryngwyn head teacher, said they run small groups in a room which showcases pupils' work and progress and provides a motivational setting.

"It's viewed very positively by both pupils and parents.

"Pupils are taking a great pride in what they're achieving and they're also enjoying it. They're encouraged to evaluate their own progress and indentify perhaps those areas where things maybe are moving a little too fast."

Maths mates

Inspectors also praised other examples of good practice, such as the use of numeracy "buddies" at Hawarden High School in Flintshire, with older pupils being trained to help coach younger pupils.

Ysgol Bryn Elian, Conwy runs a summer school for primary school pupils before they join.

Ann Keane, chief inspector of education and training at Estyn said: "Research shows that 53% of adults in Wales have numeracy skills below the level expected of an 11-year-old.

"This means they have difficulties with percentages, fractions and calculations.

"It is vital that schools continue to raise standards in numeracy by providing pupils with opportunities to develop their skills in a real life context in order to ensure they do not struggle once they get jobs and manage their own finances."
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