In this video, we introduce the concept of number 5. Same lesson plan, as used in this video, can be used for other numbers. A sketch (essential steps, sequence, questioning, and the principles of pedagogy) of this lesson is available (free of cost) by sending an email to: mahesh@mathematicsforall.org.

For the full treatment of teaching number concept watch the video: How to Teach Number Concept. Future videos will deal with numbersense and numeracy. In my opinion, almost every elementary teacher is a better reading teacher than a numeracy teacher. However, using the same principles, same care, and same teaching skills can deliver an effective numeracy lesson. The only requirement is to understand number concept mastery in the same way we expect them to know the letter concept.

As we expect every child to read fluently with comprehension by the end of third grade, we should expect and work diligently toward the goal of for every child to have mastery of numeracy with understanding by the end of fourth grade so that they can learn mathematics easily, effectively, and efficiently and appreciate its reach, power, and beauty. It is about number concept, numbersense and building the brain for numeracy and beyond. it is all about how we make it easier for students to understand it and communicate it. Through a systematic way of learning, practicing, and applying knowledge about the number concept, numbersense, numeracy, and mathematical way of thinking children have access to higher mathematics..

The videos on this site are to help achieve this goal. Join me in reaching this goal, not just in our country, but the whole world. Numeracy is the new literacy. Mahesh For a list of publications from the Center and my ideas about learning and teaching mathematics, understanding learning problems in mathematics, please visit my website http://www.mathematicsforall.org and my blog: http://www.mathlanguage.wordpress.com.

I am so thankful to my friend and colleague, Ann Lyle, for her expertise in photography. She is an accomplished, award winning, and professionally recognized photographer.

]]>**One-day Workshop**

**October 11, 2019**

**Professional Development Series**

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Several professional national and international groups, the *National Mathematics Advisory Panel *and the *Institute for Educational Sciences*, in particular, have concluded that all students can learn mathematics and most can succeed through Algebra 2. However, the abstractness and complexity of algebraic concepts and missing precursor skills and understandings—number conceptualization, arithmetic facts, place value, fractions, and integers make new learning overwhelming for many students and teachers to teach.

Some students have difficulty in mathematics because of their learner characteristics—neuropsychological and cognitive profile, poor linguistic skills, lack of prerequisite skills for learning mathematics, learning difficulties/ disabilities, such as, dyscalculia, dyslexia, and/or dysgraphia. While some others may have difficulty learning mathematics due to factors contributed by socio-cultural environmental conditions. One of the outcomes of these factors is poor number concept, numbersense, and numeracy.

Clearly, there exists a need for instruction and interventions that go beyond “typical” classroom instruction. These interventions should be effective, efficient, and elegant (that can be generalized and extrapolated). They must be based on sound principles of learning mathematics, reflecting the characteristics of the difficulty and focused on the practices that deliver outcomes envisioned. They cannot be based on just modifying the content by diluting the standards. The goal is not just being proficient in applying numeracy skills in routine situations (that can be delivered by mindless use of technology).

Being proficient at arithmetic/numeracy skills is certainly a great asset (a necessary, but not a sufficient condition) when we reach algebra; however, how we achieve that proficiency can also matter a great deal. The criteria for ** mastery**, that

(a) ** understanding** (e.g., have appropriate language and possess efficient and effective strategies, based on authentic conceptual schemas),

(b) ** fluency **(at acceptable standards), and

(c) ** applicability **(can apply to other concepts, procedures, and in problem solving).

Such a level of mastery ensures that students form strong, secure, and developmentally appropriate numeracy foundations for learning algebraic concepts and procedures so that these students can learn easier and go higher. The development of those foundations is assured if we implement the **Standards of Mathematics Practices** (**SMP**) along with the **CCSS-M** content standards.

As we expect every child to read fluently with comprehension by the end of third grade (approximately age 9), we should expect and work for every child to have mastery of numeracy by the end of fourth grade (approximately age 10) so that they can learn mathematics easily, effectively, and efficiently. A fluent reader with comprehension can apply his/her reading skills to reading content in any discipline and in any context. Similarly, a student with fluent numeracy skills (language, understanding, fluency, and applicability) should be able to apply those skills for learning algebra and model problems in intra-mathematical, interdisciplinary, and extracurricular settings.

A nation-wide discussion, verging on a political fight, is going on right now, concerning pro and con of the *Common Core State Standards in Math* (CCSS-M) and it involves every school in the country, whether they have adopted CCSS-M or not. This discussion has international and long-lasting implications for mathematics education. Even those who have not adopted them, have given considerable thought to these standards. As we see it, the implementation of CCSS-M has considerably upped the ante in mastery, rigor, coherence, and developmental trajectories of mathematics ideas. But, they have also opened the option of introducing ‘modeling,’ of realistic problems in the curriculum and teaching. That means the mathematics we teach should relate to other disciplines (i.e., STEM) and the real world (i.e., careers and professions). As a result, all school mathematics can be enlivened and made relevant, even exciting, to students by dipping into the vast array of applications (e.g., *intra-mathematical, interdisciplinary, and extra-curricular*) that mathematics has to real life. *Our message, therefore, is: math, properly taught, need not turn away our students from “good” mathematics.*

This year, in this series of workshops on mathematics education, we will cover content and strategies related to key developmental milestones in mathematics—topics that form the backbone of school curriculum. The objective is to help you learn about the content and instructional practices for teaching mathematics to all students effectively, including, those who struggle with the critical concepts and skills necessary for mastering numeracy and success in algebra.

This session will provide specific strategies and recommendations for content, instruction, intervention, remediation and draw upon currently available research-based evidence for teaching mathematics. For more information, one can go to the many posts on this blog relevant to this topic.

The topics, in this workshop, deal with understanding the issue of learning problems in mathematics, including dyscalculia and mathematics difficulties due to dyslexia, dysgraphia, and other language related difficulties from Kindergarten through high school.

In the field of dyscalculia, we have arrived at a place, where we need to move on from just focusing on the characteristics and profiles of students with dyscalculia to

*what are the reasons for the incidence and conditions of dyscalculia, and**what to do to help students who show such characteristics.*

In other words, on one hand, we should focus on understanding the nature of mathematics instruction that responds to these students’ learning needs. On the other hand, we need to focus on providing the classroom instruction so that the emergence of conditions due to dyscalculia are reduced significantly (i.e., preventive teaching). We also need to create, the classroom instruction that is dyscalculia friendly environment that minimizes the impact of conditions that may exacerbate the risk factors for dyscalculia.

In most research on dyscalculia suggestions given for instruction are too fragmented and compartmentalized. We need to present a conceptual framework that can help mathematics education professionals—classroom teachers, special educators, and interventionists better understand what their students are facing as they learn mathematics systems: language, concepts, procedures, and skills. In this area, we can learn a great deal from the progress made in the science of reading that provides suggestions for instruction and remediation for dyslexic students.

Several national and international panels have recommended instructional components for improving mathematics outcomes but presented these instructional components as a list without explicitly addressing their interrelations with learning needs, either in terms of instruction or cognitive development. We need to explore the key cognitive capacities underlying learning and conceptualizing specific mathematics ideas that specify the relationships between them. The central objective of this presentation is to provide help to classroom teachers and intervention specialists achieve better outcomes. In other words, what cognitive capacities undergird learning mathematics skills, particularly, that provide the basis for developing numeracy skills in all children. The suggestions, here, are intended to help teach the content and also improve the underlying cognitive capacities, such as executive functions: working memory, inhibition control, organization, and flexibility of thought.

*“Whether you think you can, or you think you can’t, you’re right.”*

— Henry Ford

*A. Introduction*

I have been working with people who demonstrate difficulty learning mathematics or are gifted and talented in mathematics. In this work, I meet people who have direct professional interest in the topic and others who are indirectly affected by it. People range from children age 3 to high school students to doctors and university professors and school administrators. These people raise many and diverse questions and issues to tackle problems related to mathematics learning, teaching, assessing, and organizing instruction, both classroom instruction and interventions.

Many just want to know what is “*dyscalculia*?” Some of them are looking for specific definition of dyscalculia, its symptoms, its causes, and information about protocols for diagnosis and treatment of mathematics learning difficulties, particularly dyscalculia. School administrators seek procedural advice about—what are their responsibilities, what programs and resources can help students with difficulties in mathematics. Parents search for advice on school issues related to math learning, testing, enrichment, and remediation of these issues—meeting the needs of their children.

Students, on the other hand, seek survival skills, relief from troubling math failures, and concessions and accommodations from instructors and institutions. Some of them want to learn mathematics that excites them and challenges their abilities. Many adults, who achieve success in other areas of their lives, wonder why they need to seek remedial and coping strategies to overcome baffling and frustrating conditions in learning and applying mathematics. Almost all dyscalculics seek vindication of their intelligence, and illumination and understanding of this disability.

Many students have difficulty in learning mathematics for a variety of reasons. Individuals having difficulties in learning mathematics manifest the symptoms in varying degrees and forms. One of these forms is known as dyscalculia. Not all students having difficulty in learning mathematics have dyscalculia. However, there are some basic areas of mathematical activity in everyday life that may indicate a dyscalculic tendency. That is if the mathematical activities are persistently difficult and frustrating for the person. Such symptoms may manifest as: mathematics anxiety and dyscalculia. The observations and research have shown that dyscalculic individuals are troubled by even the simplest numerical tasks such as selecting the larger of two numbers or estimating the number of objects in a display, without counting.

Dyscalculia is a lessor-known of learning disabilities that affects learners. Dyscalculia is a specific learning difficulty in mathematics. Dyscalculia is the name given to the condition that affects our ability to acquire arithmetical/ numeracy skills. People who are anxious and afraid about all things mathematical and have difficulty learning it have many other symptoms and characteristics. Most people take these coexisting conditions as “the dyscalculia syndrome.”

**B. History**

Schools have long experience of supporting children who experience difficulties with mathematics, but dyscalculia has only recently been identified as a distinct condition for children and adults. It is a fairly new term to many people. It also means that there are many adults and children who have never had their difficulties with mathematics formally identified. Furthermore, while there is currently a great deal of interest in dyscalculia in educational circles, yet there is limited body of research in this area. To date, research on math disability (MD) is far less extensive than research on reading disability (RD). Yet, like RD, MD is a significant obstacle to academic achievement for many children. There is a need to better understand its nature, its causes, and its manifestations.

While provision is made to accommodate the needs of pupils with learning problems into the school curriculum, assessment is very largely based on reading difficulties and many times the diagnosis of mathematical problems are overlooked. This is perhaps not surprising in view of the relative scarcity of information about mathematical learning problems. Although, it is becoming a focus of education, most neuro-psychological, and neurological research is concerned with understanding the basic mental processes and their role in mathematical cognition.

Other research examines the impact of traumatic brain injury on adults and children’s loss of mathematical abilities, although. The resultant mathematical difficulties is called *acalculia*. Many more people have difficulty in learning mathematics than due to mathematics learning problems or any disabilities. Many of their difficulties are not due to the conditions of learning disabilities. But, many of them assume the presence of due to some learning disability. However, they exhibit the same kinds of symptoms as dyscalculia. We term them as learning mathematics problems due to environmental factors and call them as acquired dyscalculia. *Acquired dyscalculia* becomes evident when a student, otherwise able and without learning disability, because of environmental factors—poor standards, poor teaching, lack of practice, frequent and excessive absences from school, etc., shows similar symptoms as dyscalculia.

While these research directions are increasing knowledge of the development of basic arithmetical skills (counting, addition, subtraction, multiplication and division) and of their epistemological relationships, little research has as yet explore the development of effective and efficient strategies for instruction, intervention, and remediation of fundamental skills underlying the difficulty in learning number concept, numbersense, numeracy and the development of more sophisticated domains such as algebra and geometry. The process of understanding mathematical learning problems is still in its infancy.

**C. Learning Problems and ****Mathematics Learning difficulties**

In general, the factors responsible for mathematics learning problems fall in the following categories:

- Cognitive/Neurological
- Intellectual
- Perceptual
- Language related
- Pre-requisite skills related

**Nature of Mathematics Learning Problems **

The learning problems in mathematics can be popularly categorized as:

- Developmental mathematics learning problems
- Carryover mathematical learning problems
- Math anxiety

~ Specific math anxiety

~ Global math anxiety

4. Acalculia, dyscalculia, anarithmetia, dysgraphia

5. Dyslexia and other language related mathematics difficulties

6. Acquired Dyscalculia

** Developmental mathematics learning problems** are those where the learner’s preparation for mathematics is not adequate for some developmental cognitive factors. They have difficulty in acquiring the key developmental ‘milestones’ in mathematics learning—

** Carryover problems, **on the other hand, are those where the person has difficulty in areas other than mathematics but the difficulty may interfere learning and functioning in mathematics. The difficulties may relate to language (conceptualization word problems, communication, etc.), psychomotoric problems (handwriting, spatial orientation), and emotional problems (anxiety, fear of failure, etc.).

**Environmental** learning difficulties in mathematics (acquired dyscalculia) may be the result of unsatisfactory teaching of basic concepts or of negative social influences on a pupil’s learning.

**1. Definition of Dyscalculia**

Dyscalculia is a term used to describe mathematical learning difficulties. As we know more about how children learn mathematics, why learning problems occur, and how to teach them, we know more about the nature of dyscalculia. In general, dyscalculia or acquired dyscalculia means having: intellectual functioning that falls within or above the normal range and a significant discrepancy between his/her age and mathematics skills (usually two years or more).

To be diagnosed with dyscalculia, it is important to make sure that mathematics deficits are not related to issues like inadequate instruction, cultural differences, mental retardation, physical illness, or problems with vision and hearing. It is not as commonly diagnosed as dyslexia in schools because of the lack of any strict or measurable criteria. At present, the diagnosis is by neuro-psycholgists, neurologists, or specialists in dyscalculia.

The definition of dyscalculia is, thus, evolving. In 1968, Dr. Ladislav Kosc, a pioneer in the study of mathematical learning difficulties, defined dyscalculia as follows:

Developmental dyscalculia is a structural disorder of mathematical abilities which has its origin in those parts of the brain that are anatomico-physiological substrate responsible for the maturation of mathematical abilities adequate to age without, however, having as a consequence a disorder of general mental functions. The origin may be either genetic or acquired in prenatal development. (Kosc, 1986, p. 48-49)

While this definition describes the possible causes of developmental dyscalculia, the destructive impact of poor environment (e.g., acquired dyscalculia) should not be overlooked. We need to also focus on the nurturing role played in this by appropriate education—efficient strategies, effective models and timely interventions. Whatever the cause, the effects will fit into a spectrum of problems detrimental to a child’s schooling.

There are rigorous criteria used to determine if a student has a learning disability based on and guided by special education criteria. When a student’s mathematics difficulties are severe enough to meet that criteria, special education services are indicated. However, dyscalculia has no clearly defined criteria and cannot be assessed reliably, at present. By some educational specialists, a student with any degree of mathematics difficulty may be considered to have dyscalculia. Because of the ambiguity of categorization, being identified as having dyscalculia may or may not indicate whether special education services are warranted. Nevertheless, substantial number students suffer either from dyscalculia or acquired dyscalculia.

The term learning disabilities is often misused and sometime applied, incorrectly, to students who learn mathematics in different ways or have difficulty learning mathematics. Learning disabilities in mathematics is a generic term that refers to a heterogeneous group of disorders manifested by significant difficulties in learning, acquisition and use of mathematics—reading mathematics text, writing mathematics expressions, reasoning about concepts and procedures, mathematical thinking—seeing patterns and relationships between ideas and concepts, and mastering key developmental concepts in mathematics. Mathematics is not a unary concept or skill; it is complex with multiplicity of concepts, procedures, and branches. Therefore, learning difficulties, learning problems, or learning disabilities span a spectrum.

Many of the disorders related to mathematics may be intrinsic to the individual and presumed to be due to central nervous system dysfunction. A mathematics learning disability may occur concomitantly with other handicapping conditions such as sensory impairment, mental retardation, social and emotional disturbance. It may occur along with socio-environmental influences such as cultural differences, insufficient or inappropriate instruction, or psychogenic factors, or with attention deficit disorder, all of which may cause learning problems, but a learning disability is not the direct result of those conditions or influences. Dyscalculia, as a mathematics disability, may result from neurological dysfunction and can be as complex and damaging as a reading disability, which tends to be more routinely diagnosed.

Adults with dyscalculia experience various debilitating problems in handling daily quantitative functions. The difficulty is manifested in conceptual understanding, counting sequences (skip counting forward and backward by 1, 2, 5, 10s), written number symbol systems, the language of math, basic number facts, procedural steps of computation, application of arithmetic skills, and problem solving. Mathematics learning disabilities, because of the complexity and diversity of concepts and procedures, do not often occur with clarity and simplicity. Rather they can be combinations of difficulties, which may include language processing problems, visual spatial confusion, memory and sequence difficulties, and or unusually high anxiety.

Dyscalculia is an individual’s difficulty in *conceptualizing number concept, number relationships, numbersense (intuitive grasp of numbers) and outcomes of numerical operations*. In this sense, dyscalculia only refers to issues with learning numeracy skills. Dyscalculic children may have difficulty in mastering arithmetic facts, concepts, and procedures (addition, subtraction, multiplication, and division) by the usual methods of teaching arithmetic, particularly, those that are based on counting strategies.

Dyscalculia affects an individual’s ability to estimate – what to expect as an outcome of a numerical operation and the range of answers. This difficulty manifests in a person having difficulty with estimating time, distance, and money transactions—balancing a checkbook, making change, and tipping. In other words, wherever quantity—number and calculations are involved in daily-to-day living.

Although it may be co-morbid with other difficulties, dyscalculia relates specifically to problems of mathematical language, concepts and procedure. Pupils are assessed as dyscalculic if their mathematical ability is significantly below their overall cognitive profile as determined by tests such as, the Wechsler Intelligence Scales or similar other cognitive assessments. If cognitive abilities in general are significantly below average the child is likely to be considered as having multiple, rather than specific, learning difficulties and is not dyscalculic.

**2. Types of Dyscalculia**

Dyscalculia can be broken down into four sub-types:

**Quantitative dyscalculia**, a deficit in the skills related to numeracy (e.g., computational skills—counting and calculating).**Qualitative dyscalculia**, a result of difficulties in comprehension of instructions or the failure to master the skills, symbols, and concepts (e.g., which is a difficulty in the conceptualizing of math processes)**Mixed dyscalculia**involves the inability to operate with symbols, shapes, and numbers.**Acalculia**is difficulty in learning mathematics after an insult or injury to the brain. The person had intact mathematics skills, but has lost some or many of them after the injury. The specific nature of difficulty depends on the focal area and the extent of the injury. Acalculics show the similar symptoms as the dyscalculics.

Mathematical calculations are a complex system of skills, concepts, and processes. The understanding, acquisition, and competence depends on the interaction of many abilities and cognitive mechanisms. Dyscalculic pupils are unable to use these (integrating basic mathematics skills and cognitive skills) efficiently and effectively to arrive at the solution of a problem. Many mechanisms – such as those for sequencing and organizing information – are also shared with other non-mathematical processes; consequently developmental dyscalculia frequently accompanies other learning difficulties arising from poor ** executive functions**:

(e.g., short-term–reception, working–manipulation, and long-term memory—retention);*Memory problems*(difficulty maintaining concentration, and focusing on the appropriate concept, procedure, or skill),*Inhibition control*(visual-spatial confusion, lack of organization in physical space, working equipment, or ideas, skills, and working scripts, etc.),*Organization*(rigidity in using only a limited strategy—for example, using counting as the only means to derive facts)*Flexibility of thought*difficulties (e.g., the cumulative nature of mathematics calls for heavy demands on processing information), and,*Information processing*(graphomotoric and pscho-motoric issues—poor drawings, writings, lack of clarity in executing procedures)*Motor disabilities–dysgraphia**.*

The natural anxiety of a person may also affect one’s attitude about mathematics and the resultant math anxiety from mathematics failures, in turn, affects further learning in mathematics and further complicates the picture.

Many people might relate mathematics mostly to numeracy and arithmetic; which is just a small part of a range of widely different concepts constituting mathematical knowledge. There is little in common between rote-learnt multiplication tables, the perspective geometry, coordinate geometry, probability, statistics, or calculus, for example, but they are all aspects of mathematics. Learning difficulties in these areas might be expected to manifest themselves in very different ways. Dyscalculia only affects numeracy skills and the ability to apply numerical competence in other areas of mathematics and other scientific disciplines.

Dyscalculia may affect just a few skills supporting one, or several, of these branches. For example, successful arithmetic competence requires a sound conceptual grasp of the concept and properties of numbers, their relationships underpinning computations, the decimal system, and arithmetical procedures. These are concepts that are taught to the child. They themselves are built on to other concepts that the child has been taught. Many of the cognitive processes involved in mathematical thought may also serve other non-mathematical purposes. The strengths and weaknesses of ability in each of these areas reflect our developmental history; no one will have the same combination of abilities as anybody else. Mathematics learning is woven into the fabric of the individual and then as related to the complex of mathematics system—language, concepts, procedures, and skills, in different aspects of mathematics. Consequently, mathematics difficulty can be defined with precision. Dyscalculia cannot be defined as a specific difficulty with a clearly identifiable cause or effect, if we do not limit it to the disorder of number concept, numbersense, and numeracy skills.

There are many challenges facing students with mathematical learning difficulties. They may arise from many causes, take many forms and be accompanied by other difficulties which also require intervention and remediation. For example reading difficulties may mask or accentuate accompanying mathematical difficulties; consequently intervention must address both areas to be most effective.

**D. Underlying Causes of Dyscalculia**

In our technological society mathematical ability is a valuable asset. From many perspectives, numerical skills are considered to be more important than reading abilities as a factor determining employability and wage levels and possible professional fields. The interrelationships between mathematical and other learning difficulties lead some authorities to wonder whether dyscalculia is not in fact a symptom of other difficulties, such as dyslexia and dyspraxia. This could have implications for assessment and intervention.

To understand dyscalculia better we need to look at a possible model of the mental mechanisms and processes involved in early mathematical learning, particularly, the number concept, numbersense, and numeracy as the competence in these areas is the basis of dyscalculia.

Dyscalculia has several underlying causes. One of the most prominent is a weakness in ** visual processing **and

Another problem is with ** sequencing**. Students who have difficulty with sequencing or in organizing detailed information often have difficulty remembering specific facts and formulas for completing their mathematical calculations, particularly, procedural calculations.

Like dyslexia, dyscalculia can be caused by a visual perceptual deficit. Along with dyslexia, the extent to which one can be affected varies uniquely with the individual. Like dyslexia there is no single set of signs that characterize all dyslexics, there is no one cause of dyscalculia. However, dyscalculia refers specifically to the inability to perform operations in mathematics or arithmetic.

There may be more students with and without learning disabilities in any mathematics class, who have problems or difficulty in learning mathematics than we realize. If a class has learners who read numbers backwards, have trouble telling time, confuse part–whole relationships, have difficulty keeping score in a game, and have difficulty remembering arithmetic facts, ideas behind key concepts, strategies/rules in basic operations and formulas, and sequence of steps in key arithmetic procedures, they may be learning disabled.

Everyone forgets occasionally, but when learning every concept is difficult and the student consistently forgets it, it is a symptom of disability and calls for intervention. According to the National Adult Literacy and Learning Disabilities Center, “it is estimated that 50 percent to 80 percent of students in Adult Basic Education and literacy programs are affected by these learning disabilities,” (1995, p. 1). Some of these have dyscalculia. However, many of them have acquired dyscalculia. The implications of such a staggering statistic for the adult basic education (ABE) teacher are worth further investigation. However, those with specific learning difficulty, dyscalculia, even in this group are much smaller. In school age children it is much smaller. Chinn and Ashcroft (1997) report that from a sample of 1200 children only 18(1.5%) had purely mathematics specific learning difficulties. Many of them, however, will develop acquired dyscalculia, if effective, efficient strategies are not taught and emphasized in regular mathematics instruction.

Since people are just becoming aware of this condition, it is hard to quantify exactly how many people have dyscalculia. Although many people experience difficulty or disability in mathematics, some of the recent studies show that dyscalculia –difficulty with numbers and number operations—afflicts between 5% and 6% of the population, based on the proportion of children who have special difficulty with mathematics despite good performance in other subjects. After a long period of growing awareness it is now widely accepted that dyslexia affects a significant proportion of the population and provisions are made to facilitate their situation as far as possible. Awareness of dyscalculia, however, is far lower although it may be at least as common and as far reaching as dyslexia, longitudinal studies in Europe, Israel and the USA suggest the same—that is about 5-6% of the population are affected by some degree of dyscalculia. However, the proportion of the population with purely mathematical difficulties may be far lower.

In very simple terms, analogous to dyslexia—where the dysfunction manifests in difficulties in reception, comprehension, or production of linguistic information, dyscalculia can be defined as the dysfunction in the reception, comprehension, or production of quantitative and spatial information. However, dyslexia may also affect learning mathematics. Dyslexics frequently have difficulties with certain areas of mathematics; according to the British Dyslexia Association (1982) approximately 40 to 60% of dyslexics experience some mathematical difficulty.

2. **How does dyscalculia develop?**

Schools have supported children who experience difficulties with mathematics, but dyscalculia has only recently been identified as a distinct condition for children and adults. It means that there are many adults and children who have never had their difficulties with mathematics formally identified.

Our work with children and adults with learning problems in mathematics suggests that there seem to be several factors that may be implicated as the causes of mathematics learning problems:

- Cognitive factors,
- Inadequate and poor teaching—mismatch between mathematics learning personality of a student and teaching style,
- Lack of pre-requisite skills for mathematics learning,
- Delay in the development of mathematics language—vocabulary, syntax, and translation ability from mathematics to English and English to mathematics,
- Inadequate mastery at levels of knowing: movement from intuitive to concrete, concrete to representational, representational to abstract, abstract to applications, and from applications to communication.

In most cases of dyscalculics, it seems, the pre-requisite skills for mathematics learning are affected. These prerequisite skills include: following sequential directions, spatial orientation/space organization, pattern recognition, visualization, estimation, inductive and deductive thinking. These prerequisite skills act as “anchors” for mathematics ideas. The degree to which these prerequisite skills are not developed or affected varies from learner to learner.

There is some correlational evidence between the co-incidence of dyslexia and dyscalculia. But a clear link between dyslexia and dyscalculia hasn’t been proved. The International Dyslexia Association has suggested that 60% of dyslexics have some difficulty with numbers or number relationships. Of the 40% of dyslexics who don’t have mathematics difficulties, about 11% excelled in mathematics. The remaining 29% have the same mathematical abilities as those who don’t have learning difficulties. Many dyslexia specialists believe that for many dyslexic people the difficulties, which affect their reading, and spelling also, cause problems with mathematics.

Since some of the same pre-requisite skills are involved in both language acquisition and mathematics – at least in the early learning concepts and grade levels – the coincidence of dyslexia and dyscalculia is not uncommon. Our observations show that about 40% of dyslexics also exhibit some symptoms of dyscalculia. However, the group of dyscalculic children/adults, like the group of dyslexics, is not a homogeneous one. Most people with dyscalculia don’t necessarily suffer from any other learning difficulty. Indeed, they may well excel in non-mathematical areas.

All mathematics teachers have encountered children with mathematics learning difficulties and varying degrees of mathematics anxiety. Most of these teachers have some awareness of the nature of learning disabilities/problems in mathematics. However, few teachers are aware of the causes of these problems—learning disabilities, mathematics anxiety, and dyscalculia. In fact, very few of them are able to recognize and deal with the problems of dyscalculics.

American Academies of Neurology and Pediatrics have identified dyscalculia as one of the neurological conditions with a cluster of syndromes associated with it. Similarly, in 2001, as part of the national Numeracy Strategy in the UK, the government published guidance for teachers to provide classroom help to support dyscalculic pupils. Dyscalculia is likely to be a more familiar condition to people who specialize in learning difficulties such as special needs coordinators and educational psychologists. In the U.S., many school psychologists, neurologists and neuro-psychologists have begun to diagnose this as a condition. In spite of this, the general public and teachers have limited understanding of the condition of dyscalculia. Early diagnosis of the problems, effective planning of intervention, and effective and efficient remediation support can reduce the number of struggling students in mathematics when this information becomes available to more mathematics and classroom teachers.

Many students with disabilities have histories of academic failure that contribute to the development of *learned helplessness* in mathematics. It is important that mathematics instructors recognize the symptoms of dyscalculia and take the necessary measures to help students that are affected.

**E. Mathematics Symptoms of Dyscalculia**

Symptoms of dyscalculia and other mathematics difficulties are manifested in several ways:

- Linguistic
- Cognitive/content/conceptual
- Procedural
- Behavioral (Math Anxiety—global and specific)

Many of dyscalculics students, even when they can produce a correct answer or use a correct method, they may do so mechanically, without conceptual understanding and confidence and using inefficient methods and strategies. Some of the manifested symptoms of dyscalculia are:

Dyscalculia is a collection of symptoms of learning disability involving the most basic aspect of arithmetical (quantitative and spatial) skills. On the surface, these relate to basic concepts such as: number concept number facts, estimation, telling time, calculating prices and handling change, and measuring things such as temperature and speed.

Dyscalculia is an individual’s difficulty in conceptualizing numbers, number relationships, outcomes of numerical operations and estimation – what to expect as an outcome of an operation. Math problems begin from number concept. Math disabilities, therefore, can arise at nearly any stage of a child’s scholastic development. While very little is known about the neurobiological or environmental causes of these problems, many experts attribute them to deficits in one or more of five different skill types. These deficits can exist independently of one another or can occur in combination. All can impact a child’s ability to progress in mathematics.

**(a) Incomplete Mastery of Numberness and Number Facts**

When a student above the age of 8 has to count the dots on a domino or a playing card, this shows that the student has not conceptualized number and may be a prime candidate for the identification of dyscalculia or acquired dyscalculia. A surprising number of people resort to counting to work out the simplest of quantitative tasks.

Children who have not mastered numberness are at risk for learning number facts. Numberness is the integration of one-to-one correspondence, visual clustering, sequencing, and decomposition/recomposition of quantity. Number facts are the basic computations (9 + 3 = 12 or 2 × 4 = 8) students are required to master (as defined earlier) in the earliest grades of elementary school. Recalling these facts efficiently is critical because it allows a student to approach more advanced mathematical thinking without being bogged down by simple calculations.

**(b) Computational Weakness**

Many students, despite a good understanding of mathematical concepts, are inconsistent at computing. They make errors because they misread signs or carry numbers incorrectly, or may not write numerals clearly enough or in the correct column. These students often struggle, especially in primary school, where basic computation and “right answers” are stressed. Often they end up in remedial classes, even though they might have a high level of potential for higher-level mathematical thinking. In general, they show

- Poor mental math computation ability, often fear of and difficulty in common usage of number such as in money transactions—balancing a checkbook, making change, and tipping.
- Difficulty with math processes (e.g., addition, subtraction, multiplication) and concepts (e.g., sequencing of numbers).
- Difficulty with estimation with and without calculations.
- Difficulty with rapid processing of math facts.
- Have difficulty using a calculator properly because of absence of concepts and estimation.
- Difficulty keeping score during games, or difficulty remembering how to keep score in games, like bowling, etc.

**(c) Difficulty Transferring Knowledge**

One fairly common difficulty experienced by people with math problems is the inability to easily connect the abstract or conceptual aspects of math with reality. Understanding what symbols represent in the physical world is important to how well and how easily a child will remember a concept. Holding and inspecting an equilateral triangle, for example, will be much more meaningful to a child than simply being told that the triangle is equilateral because it has three sides with equal length. And yet children with this problem find connections such as these painstaking at best.

**(d) Recognizing Patterns and Making Connections**

Some students have difficulty seeing and extending patterns, making meaningful connections within and across mathematical experiences. For instance, a student may not readily comprehend the relation between numbers and the quantities they represent. For example, realizing that adding two same numbers is same as knowing table of 2. Knowing commutative property of addition/multiplication means, you have to memorize only half as many facts. If this kind of connection is not made, math skills may be not anchored in any meaningful or relevant manner. This makes them harder to recall and apply in new situations.

**(e) Incomplete Understanding of the Language of Math**

Mathematics is a second language for most children. It has its own vocabulary, syntax and rules of translation. The vocabulary, syntax and translation from English to math and math to English may impact mathematics learning.

For someone who has trouble distinguishing letters, a + sign might be confused with ×, for example. The language of mathematics can also be a problem, they latch onto the first meaning they know for words, for example, distributive property means, “distributing.”

Therefore, for some students, their math disability is driven by problems with language. These students may also experience difficulty with reading, writing, and speaking. In math, however, their language problem is confounded by the inherently difficult terminology, some of which they hear nowhere outside of the math classroom. These students have difficulty understanding written or verbal directions or explanations, and find word problems especially difficult to translate.

A student with language problems in math may:

- Have difficulty with the vocabulary of math,
- Be confused by language in word problems and applications,
- Not know when irrelevant information is included or when information is given out of sequence,
- Have trouble learning or recalling abstract terms,
- Have difficulty understanding directions,
- Have difficulty explaining and communicating about math, including asking and answering questions,
- Have difficulty reading texts to direct their own learning,
- Have difficulty remembering assigned values or definitions in specific problems,
- It is common for students with dyscalculia to have normal or accelerated language acquisition: verbal, reading, writing, and good visual memory for the printed word, however, some have difficulty with math language, and
- Mistaken recollection of conceptual names, terms, definitions, and expressions.

**(f) Perceptual Difficulties **

It deals with** **difficulty comprehending the visual and spatial aspects of mathematics. A far less common problem—and probably the most severe—is the inability to effectively visualize math concepts. Students who have this problem may be unable to judge the relative sizes among three dissimilar objects. This disorder has obvious disadvantages, as it requires that a student rely almost entirely on rote memorization of verbal or written descriptions of math concepts that most people take for granted. Some mathematical problems also require students to combine higher-order cognition with perceptual skills, for instance, to determine what shape will result when a complex 3-D figure is rotated.

**(g) Memory Difficulties**

The memory shortcomings associated with dyslexia obviously causes problems with mental arithmetic. Even if children learn their times tables they cannot sequence backwards and forwards – so if you ask them what six times four is, they will have to start again counting through. That does not provide fluency.

Because of the mathematics curriculum and programs and their pace: If a child slips behind it became more and more difficult to catch up. Efficient strategies can make it happen, but for learning strategies, to some extent, depends on memory. The size of working memory and its effective use id critical for mathematic learning.

**(h) Difficulties due to Spatial Orientation/Space Organization**

Spatial orientation/space organization is highly correlated with mathematics achievement. When students have poor spatial skills, they have difficulty in multiple areas of mathematics. For example,

- Poor sense of direction (therefore, problems with place-value, aligning numbers, reversals, confusing different forms of numbers because of their spatial locations, etc.), easily disoriented, as well as trouble reading maps, telling time, and grappling with mechanical processes.
- Have trouble with sequence, including left/right orientation. They will read numbers out of sequence and sometimes do operations backwards. They also become confused on the sequences of past or future events. It greatly affects learning and mastering order of operations (GEMDAS—grouping, exponents, etc.)
- Poor memory for the “layout” and structure of problems and organization of their work, and geometrical designs and figures. Gets lost or disoriented easily.
- May have poor eye-hand coordination, difficulty in writing mathematics expressions and problems.

** (i) Executive Function related**:

- There is sometimes poor retention and retrieval of concepts, or an inability to maintain a consistency in grasping mathematics computation rules.
- Difficulty with abstract concepts of time and direction, schedules, keeping track of time, and the sequence of past and future events.
- Inability to grasp and remember mathematics concepts, rules formulas, sequence (order of operations), and basic addition, subtraction, multiplication and division facts. Poor long-term memory (retention and retrieval) of concept mastery. Students understand material as they are being shown it, but when they must retrieve the information, they become confused and are unable to do so. They may be able to perform mathematics operations one day, but draw a blank the next. May be able to do classroom and homework and book work but can fail all tests and quizzes.

**F. Dealing with dyscalculia: What forms of instruction are most effective? **

Dyscalculia is a special need, and requires diagnosis, support and special methods of teaching. The support should give the learners an understanding of their condition, and equip them with coping and efficient learning strategies that they can use in the classroom and in their day-to-day encounters with quantity and space. Since this is a heterogeneous group no general or single intervention can be recommended.

Dyscalculic learners lack an intuitive grasp of numbers, recognizing umber relationships, and have problems learning number facts and procedures by the usual methods of teaching. Most of their arithmetic facts are derived by counting and procedurally. Even when these learners produce a correct answer or use a correct procedural method, they may do so mechanically and without confidence; they are anxious about it.

Therefore, first objective of remedial instruction and intervention is helping them acquire efficient strategies for developing number concept. That success will improve learners’ self-esteem. The same process should be repeated for all of the developmental milestones. The developmental milestone concepts for learning mathematics are: *understanding number concept, number relationships (arithmetic facts)*, *place value (of large and small numbers)*, *fractions (including fractions, percents, ratio and proportions)*, *integers*, *spatial sense* and the *concept of variability*. Because a person’s mathematics difficulties generally originate from some dysfunction in one of these milestone concepts, intervention should begin with effective and systematic instruction in these areas. For example, a student in third grade with gaps should get extra intervention in the development and mastery of number concept and numbersense.

Individuals with dyscalculia need help in organizing and processing information related to quantity and space. Since mathematics is a form of language, one should spent time on its vocabulary and syntax and translation from mathematics to English and from English to mathematics.

These individuals can benefit from tutoring that can accomplish three objectives.

**First,** to help them make-up the missing arithmetic concepts—number concept, numbersense, and numerical operations. However, this should not be done in isolation. Whatever facts are learned (mastered) should be applied to another mathematics concept or a problem.

**Second,** to help them connect these concepts to their current mathematical needs.

And, **third**, is to help them develop the pre-requisite skills for future mathematics learning.

**1. The CPVA Model of instruction**

Effective teaching combines ** direct instruction** (teacher-directed tasks, discussion, and concrete models) with

*Intuitive*** > Concrete > Pictorial > Abstract > Applications > Communication. **

However, the most important steps in this developmental sequence of a concept or procedure are: **Concrete **modeling (that are efficient and effective) to **Pictorial representation** (efficient and effective and congruent to concrete models) to **Visualization** (picturing the model and rehearsing the script in the mind’s eye) to **Abstract **(recording it in abstract form—symbols, formula, procedure, equations, etc.) (**CPVA)**. The sequence (CPVA):

**Concrete > Pictorial > Visualization > Abstract**

is the key to reaching all children including dyscalculic and dyslexic in a mathematics class or individual setting. Example of CPVA:

**Cuisenaire rods **or** Montessori Colored rods (Concrete) > **

**Visual Cluster Cards **or** Empty Number Line (Pictorial)** >

** Imagining Visual Cluster Cards patterns and constructing** **Empty Number Line in the mind’s eye (Visualization) > **

**Recording **on paper using** mathematical symbols (Abstract).**

The concrete model helps develop the concept and language. Transfer from Concrete to pictorial helps to do the task analysis and helps create the script for implementing and understanding the task. Visualization helps reinforce the understanding and the transfer from concrete/pictorial to abstract and solidifies the script. To be effective the representations (concrete, pictorial, and abstract) should be congruent and the script from one representation to the other should be consistent. Effective questioning from the teacher and proper language helps develop the script efficiently. Such teaching include:

(breaking down the task into parts and then synthesizing the parts into a whole, providing step-by-step prompts),*Sequencing and task-analysis*(automatizing arithmetic facts, daily testing, sequenced review)*Repetition and practice*and responses (structured questioning where teacher asks process or content questions to scaffold learning and develops scripts for tasks and steps under guidance)*Socratic questioning*culty (the teacher provides necessary assistance or tasks sequenced from easy to difficult)*Control of task diffi*- Proper
(after mastering facts, concepts, and estimation)*use of technology* before collaboration activities and individual practice*Teacher-modeled problem solving*(reminders and scripts to be used in strategies)*Strategy cues*(each concept and procedure after it has been learnt should be connected to other related concepts, procedures, and skills), and*Making connections*of tasks to improve performance and self-esteem.*Frequent practice and assessment*

Every remedial/instructional intervention session should have the following components:

- Developing the prerequisite and support skills
- Learning arithmetic facts orally
- Visualizing problems and information
- Verbalizing and recording procedures and estimation
- Helping the child form problems relating to the given concept
- Counting forward beginning with a given number e.g. begin with 53 and count forward by two’s, by three’s, etc.
- Counting backwards beginning with a given number, e.g., begin with 97 and count backward, by 1’s, 2’s, 5’s, and 10’s.
- Showing patterns of number facts, e.g., 4 + 4 = 8, then 4 + 5 = ___; 8 +10 =18, then 18 + 10 = ___, 78 + 10 = ____, etc.

The three components of a mathematical idea: ** linguistic**,

2. *Vertical Acceleration*

Based on our experiences with many dyscalculics, we find that with the help of a competent tutor, with effort, discipline and structure, and appropriate concrete materials, dyscalculics can make progress in mathematics learning and realize their potential.

In the case of students who are several years behind their classmates, we can develop the concepts sequentially—we cannot keep on working on additive reasoning till everything in it is mastered, we need to connect whatever is learned in additive reasoning should be connected and transferred to multiplicative, proportional, and algebraic reason.

It should be remembered that problems of dyscalculics are related to numeracy; therefore, the development of number concept, numbersense, and numeracy is the most important part. However, one should not do it exclusively. Once one arithmetic fact is mastered, for example, the table of 4, then the student, particularly older student, should practice multi-digit number multiplied by 4 (e.g., 45678 × 4 =?), multi-digit number divided by 4 (44678 ÷ 4), fractions with numerator and denominator that are multiples of 4 (e.g., simplify the fractions* 4/20, 20/32, 2x/8x, 240/800,* etc. to the lowest term: , etc.), solve the equation: (4*x=28, 4y=56, etc.). *The choice of problems and topics is dependent on the age/grade level of the student.

This approach is called ** Vertical Acceleration**. That means start with a lower concept (start with wherever the student is) that means however so the low the student is, start there, and then move vertically. Begin with a concept, fact, procedure that the student does not know, and take the concept to higher levels. For example, multiplication procedure of whole numbers is extended to multiplication of fractions, to decimals, to integers, to variables. This is only possible, if the teacher uses efficient and effective models and materials (e.g., for multiplication, one uses area model), appropriate language, and common script and procedures. In designing instruction, it is important to keep in mind the following aspects of learner characteristics, instructional components, and the nature of the learning problem.

**3. Learner Characteristics and Learner Differences**

• Nature of mathematical problems

The role of a teacher, particularly, the interventionist is not just to help students to acquire the content, but also to help them improve their learning skills. Today, neuroscience research has found that the plasticity of the brain holds possibilities of improving learning capabilities of students in learning material that otherwise was not in the realm of possibilities. To optimize mathematics learning and for students to acquire competence, teachers should use strategies from the following areas of learning theory:

**Cognitive Strategies —**Improving children’s cognitive strategies prepares them to learn mathematics more effectively. These strategies may include the use of concrete materials and other models, inquiry techniques, metacognition, and others.**Different Ways of Learning**— Understanding that individuals may learn differently can help teachers develop appropriate instructional strategies.**Prerequisite Skills for Mathematics Learning**— Developing appropriate prerequisite skills will help to anchor students’ mathematics learning.**Learning Difficulties —**Developing an awareness of students’ mathematics learning problems–developmental mathematics learning disabilities, language acquisition, carryover problems, dyscalculia, and mathematics anxiety–will help teachers to mitigate their effects in the classroom.**Instructional Design**: The focus of the instructional activities should be developing strategies in the areas of: cognition–perceptions, executive functions, memory systems, higher order thinking; mathematical–fundamental thinking skills (decomposition/recomposition, additive and multiplicative reasoning, and mathematical way of thinking. The mathematical models and strategies should be exact, efficient, elegant (can be generalized, extrapolated, and abstracted.

**(a) Cognitive Preparation and Mathematics Conceptualization**

Methods for improving cognitive preparation are:

- Using appropriate, universal concrete models to introduce mathematics content,

- Asking many hypothetical questions when students are engaged in learning,

- Developing metacognition by connecting success to the factors that made the success possible and helping students reflect on their strategies and actions.

Appropriate universal concrete models help children develop cognitive strategies easily. During this concrete manipulative work, the teacher must pay attention to the development, articulation and application of these strategies. This means:

- It is not enough to look at a child’s answer, although answer is important.
- Examine the strategies by which s/he answers the problem
- Children always use some kind of strategy
- Sophistication of the strategy varies with the child, the concept, and familiarity with the problem.
- The more advanced the strategy, the more advanced the thinking.

Teacher’s questions are the mediating link between instructional models and the content that students acquire. The quality and the proper sequencing of these questions determine the quality and depth of learning. A question sets off a sequence of cognitive functions, In fact:

*Questions instigate language;**Language instigates models;**Models instigate thinking;**Thinking produces understanding;**Understanding results in competent performance;**Competent performance produces long lasting self-esteem; and**Self-esteem is the basis of meaningful learning.*

Nature (quantity and quality) of teachers’ questions determines the nature and quality of children’s learning and achievement. Convergent questions produce very little language; divergent questions produce more language and learning.

Choice of instruction models, type and frequency of questions asked, the level of language used and expected, sequence of tasks selected and designed, and the form, variety, and frequency of assessment determine the effectiveness of the teacher.

**Mathematics Learning Personality: Ways of Making Sense of Mathematics Information**

Each one of us makes sense of mathematics information uniquely using strategies and approaches indicating preference for language, concepts, and procedures. It is, therefore, important to pay attention students’ mathematics learning personalities and ways of making sense of mathematics information.

Matching teaching approach, conceptual models, and nature of language usage with a student’s mathematics personality results in his/her learning that is easier, deeper, and more productive.

- Following Sequential Directions
- Spatial orientation/space organization
- Pattern recognition and its extension
- Visualization
- Estimation
- Deductive reasoning
- Inductive reasoning

2. Teaching Style and Roles: Methods of Communicating the Content and delivering instruction

3. Empathy with the Learner and the Content

4. Teaching models

PART II

Numbersense: A Window to Understanding Dyscalculia

Acquiring the *number concept *or* numberness*—understanding number, its representation, and its applications, is a fundamental skill. It is like acquiring the alphabet of the mathematics language with arithmetic facts as its words.

Much of the research (Geary, 1993; Robinson et al., 2002) has focused on developing a theoretical understanding of mathematics learning difficulties. This article looks at the role of number concept and numbersense in mathematics learning difficulties and implications for instruction and interventions. Children’s understanding and level of mastery of number concept and numbersense provides a window into their arithmetic difficulties, particularly dyscalculia Dehaene et al., 1998; 1999; Gersten & Chard, 1999).

Numbersense deals with number concept, number combinations—arithmetic fact, computing and place value. Numbersense is a cluster of integrative skills: number concept, making meaning and ways of representing and establishing relationships among numbers, visualizing the relative magnitude of collections, estimating numerical outcomes, and mastering arithmetic facts and proficiency in their usage (Dehaene et al., 1999; Fleischner et al., 1982). Numbersense is the flexible* *use of number relationships and making sense of numerical information in various contexts. Students with numbersense can represent and use a number in multiple ways depending on the context and purpose. In computations and operations, they can decompose and recompose numbers with ease and fluency. This proficiency and fluency in numbersense helps children acquire *numeracy***.**

*Numeracy* is the ability to execute standard whole number operations/ algorithms correctly, consistently, and fluently with understanding and estimate, calculate accurately and efficiently, both mentally and on paper using a range of calculation strategies and means. Numeracy is the gateway to higher mathematics beginning with the study of algebra and geometry.

Many individuals encounter difficulties in mastering numeracy. Some because of (a) environmental factors—lack of appropriate number experiences, ineffective instruction and a fragmented curriculum, inefficient conceptual models and strategies, lack of appropriate skill development, and low expectations, and (b) individual capacities and learning disabilities. For example, teaching arithmetic facts by sequential counting (“counting up” for addition, “counting down” for subtraction, “skip counting” on number line for multiplication and division), as advocated by many researchers and educators, is not an efficient strategy for many children including dyscalculics (Gelman & Gallistel, 1978; Gelman & Meek, 1983; Gelman et al., 1986).

Among those who exhibit learning problems in mathematics, some experience difficulty in specific aspects of mathematics—difficulty only in procedures, in conceptual processes, or in both. Some have difficulty in arithmetic, algebra or geometry. Some may have general learning disabilities in mathematics while others display symptoms only of dyscalculia.

Learning disability may manifest as deficits in the development of prerequisite skills: following sequential directions, spatial orientation/space organization, pattern recognition and extension, visualization and visual perception, and deductive and inductive thinking. These deficits may affect learning ability in different aspects of mathematics, for example, a few isolated skills in one concept/procedure or several areas of arithmetic/mathematics. Some learning problems fall in the intersection of quantity, language, and spatial thinking.

Because of the range of mathematics disabilities, we cannot clearly identify a cause or effect; no one explanation adequately addresses the nature of learning problems in mathematics. Most mathematics problems and difficulties such as carryover problems, dyscalculia, or mathematics anxiety are manifested as lack of quantitative thinking. In this chapter, we are interested in one area of mathematics disabilities, the problems related to numeracy due to dyscalculia or acquired dyscalculia.

A. **Nature of Number Related Learning Problems: Dyscalculia**

Difficulties associated with numberness, numbersense, and numeracy are known as ** dyscalculia**. Dyscalculia has the same prevalence as dyslexia (about 6-8% of children) although it is far less widely recognized by parents and educators (Ardilla & Roselli, 2002).

Dyscalculia is manifested as poor number concept, difficulty in estimating the size and magnitude of numbers, lack of understanding and fluency in number relationships, and inefficiency of numerical operations. Dyscalculics depend on immature and inefficient strategies such as sequential counting to solve problems that most children know by heart. At the same time, they find it hard to learn and remember arithmetic facts by sequential methods. Like dyslexics, they need special academic support. When taught with appropriate methods and efficient models, children respond favorably (Cohen, 1968; Dunlap & Brennen, 1982; Shalev et al., 2005).

A characteristic many dyslexics share with dyscalculics (Light & DeFries, 1995) is limited lexical entries for number and number relationships thus facing problems with automatic labeling the outcome of number relationships—instant recall of arithmetic facts (e.g., multiplication tables). They do not have “sight facts” in their minds for numbers.

Sight facts are like sight vocabulary, for example, knowing that 7 is 6 and 1; 5 and 2; and 4 and 3. Sight facts are instrumental in achieving automatization, the fluency to produce, for example, the fact 8 + 7 = 15 in 2 seconds or less orally and 3 seconds in writing and understanding (using a non-counting strategy, e.g., 8 + 7 is one more than 7 + 7, therefore, 8 + 7 = 15, or 8 + 2 is 10 and then 5 more is 15). This lack of automatization, in most cases, is an artifact of poor instruction rather than real difficulty or disability.

Problems that most dyscalculics face in arithmetic are due to poor number conceptualization and numbersense (Dehaene, 1997). Without exposure to efficient and effective methods of learning, children do not acquire proper number concepts, arithmetic facts, and standard procedures and risk not gaining proficiency in mathematics by the end of first grade. Lack of success in the development of number becomes the main reason for a child’s difficulty in learning mathematics and dyscalculia (Jordan, Hanich et al. as cited in Gersten et al., 2005).

Just as it is possible to build lexical entries for words, letters, and word-parts, it is also possible to acquire strategies to develop lexical entries for numbers, numbers facts, symbols, formulas, and even equations. Although mathematical symbols themselves are not phonetic, each symbol represents a lexical entry whose meaning and interpretation can be understood (Ball & Blanchman, 1991).

B. **Literacy and Numeracy**

There are many parallels in the development of literacy and numeracy, which we need to explore. Young children develop literacy through literacy practices (e.g., being read to at bedtime). Similarly, early exposure to the language and symbols of quantity and space creates lexical entries for quantity (number words) and the role of number (size/quantity)— what and how to quantify, what and how to measure, and how to represent and use quantities (Adams, 1990).

The complex process of mastering reading involves a variety of brain components and systems—both localized and global—that perform and integrate tasks such as recognizing and organizing symbols—visual and aural, discerning and analyzing sound patterns, perceiving spatial arrangements—source of speech or location of the symbol, and verbal and non-verbal clues. Some of the same mechanisms are called upon in acquiring numeracy and are related to language, visuo-spatial, sequencing, and working memory. A breakdown and deficits in any of these areas may affect learning letters and numbers alike. Many dyslexics, therefore, show symptoms of dyscalculia (Light & DeFries, 1995).

While there are important similarities in learning to read and conceptualizing number, there are also important differences. Some unique abilities and systems are needed to learn number and its applications. For this reason, there are people who can read and have poor numeracy skills, but there are very few numerates who cannot learn to read. Keeping in mind these unique differences, we need to design activities for making numeracy accessible to all children.

**1. Phonemic Awareness, Numberness, and Numbersense**

Fluent reading and fluency in numberness are analogous. Research (Williams, 1995) in reading shows that phonemic awareness—the insight that words are composed of sounds and the ability to connect fluently grapheme to phoneme and phonological sensitivity—the ability to break words into meaningful “chunks” and then “blend” them fluently—are predictors of early reading performance (better than IQ tests, readiness scores, or socioeconomic level) and essential for reading acquisition. Processes of numberness—one-to-one correspondence, sequencing, visual clustering, and decomposition/re-composition, representation of number orally and graphically— are similar. The ability to associate a number to a cluster is like phonemic awareness and the ability to instantly recognize that a number is made of smaller numbers (decomposition/ recomposition) is equivalent to chunking and blending. Numberness is a predictor for future proficiency and fluency in arithmetic.

Understanding of phonemic awareness has revolutionized the teaching of beginning reading. Numberness and numbersense carry similar implications for instruction for children with or without learning difficulties. The proper definition, and development of numberness and numbersense is the key to planning remediation for dyscalculics and preventing acquired dyscalculia. However, educators and psychologists have taken a narrow view of number concept—e.g., ability to count forward and backward (Gelman & Gallistel, 1978; Klein, Starkey, & Ramirez, 2002; McCloskey & Mancuso, 1995; Moomaw & Hieronymous, 1995).

In reading, one needs to focus on the phonemes in a word; in math, one needs to see clusters of objects in the mind’s eye. Most children have difficulty forming visual clusters in their minds and sight facts by one-to-one counting. Decoding letters in a word does not lead to reading; similarly, counting individual objects/numbers (concretely or sub-vocally) does not lead to numberness. In fact, one-to-one counting turns most children into counters – that’s all. To conceptualize number, one needs to see clusters (decomposition) in a collection and integrate smaller clusters into larger clusters (recomposition). Associating a number name to the collection and relating this number to smaller clusters (numbers) is forming sight facts. Recognizing clusters (sight facts) is like recognizing phonemes and sight words. With the help of sight facts, children can move beyond counting and learn arithmetic facts at an automatized level. Many LD children have difficulty forming visual clusters in their minds and sight facts by one-to-one counting (Schaeffer et al., 1974).

The mastery of numberness and proficiency in arithmetic and phonemic awareness and the ability to read with understanding are parallel activities, nevertheless, it is also important to recognize the differences between the two processes. Phonemic awareness involves focus on auditory processes and phonological decoding associates grapheme and phoneme, whereas visual perceptual integration—recognizing clusters, estimating by observation, and decomposition/recomposition of clusters—is fundamental to the development of numberness.

**C. Parallels: Letter Recognition and Number Concept—Numberness **A child knows the alphabet when he can

- Identify the letter (shown M, he recognizes it instantly),
- Recognize the letter in its variant forms (e.g.,
**M**, M,**M**,**M**,**M**, M, M, M, m. etc.), - Recognize letters among other symbols (e.g., M in CALM, MILK, WARMER, $M$569A, etc.),
- Write the letter and describe the various strokes in the proper order, and
- Associate a sound to the letter (e.g., M as in monkey).

This should be true for all letters of the alphabet. Mastery of number is similar and is more than just reciting and writing the numbers. A child has number concept when he

- Possesses lexical entries for number (knows number names and the difference between number words and non-number words), (Fuson, 1980; Fuson et al., 1982)
- Can meaningfully count (one-to-one correspondence + sequencing), (Fuson et al., 1982; Piaget, 1968; Pufall et al., 1973; Saxe, 1979)
- Can recognize and assign a number to a collection/cluster (organized in a pattern up to ten objects) without counting (Resnick, 1993)
- Can represent a collection—a
**visual cluster**of seven objects à**graphical**representation, e.g., 7, - Can write the number when heard (hears
*s-e-v-e-n*and writes 7), and - Can decompose and recompose a cluster into two sub-clusters (i.e., a number, up to 10, as sum of two numbers and vice-versa).

Images of visual clusters in the mind’s eye provide a child a base of “sight facts.” For example, when one sees the visual cluster of 7 objects, one recognizes the sight facts: 7 = 1 + 6 = 2 + 5 = 3 + 4 without counting. These sight facts, with strategies of addition and subtraction based on decomposition/recomposition, provide a strong base for arithmetic facts mastery beyond 10.

*Numberness*, thus, is the integration of:

Mastery of number concept/numberness, arithmetic facts (arrived at by using decomposition/recomposition) and place value is called ** numbersense**. Lack of proper instruction in numberness and numbersense poses conditions of failure in early mathematics. Instruction in strategies for deriving arithmetic facts and procedures are much more productive when number concept is intact (Geary, 1994; Gelman, 1977).

**D. Language and Number**

Early number conceptualization begins with concrete experiences—counting objects in context.

Many children become and remain counters because of this early emphasis on counting. Appropriate concrete experiences accompanied with rich language, on the other hand, help abstract the experience into concepts with labels. Neither concrete experiences alone nor purely language-based teaching develop the concept of number for all children. For abstraction of concrete experiences into numberness, language is essential. Children must transcend the concrete models in order to learn to solve problems and communicate through mathematical symbols. This concurrent thinking of numbers as concrete and abstract is at the core of true number conceptualization and is a real challenge for many children (Baroody, 1992; Brainerd, 1992; Copeland, 1974).

**1. Mathematics Language and Native Language**

In the child’s native language, numbers function as predicators and qualifiers: five dishes, many books, fewer children, etc. They function like adjectives in a sentence. Most children are quite fluent in this before they enter Kindergarten (Carruthers & Wortington, 2005).

Later numbers function not only as predicators but also as real, concrete objects: six hundred is a big number, an even number, or much smaller than the number six hundred thousand. Thus, in the language of mathematics, numbers are qualifiers as well as ‘real’ abstract objects. Mathematical operations can be performed on numbers when we treat them as real, concrete entities (Williams, 1977).

Conceptualizing number requires a child to perform two simultaneous abstractions: to* translate sensory, concrete representations of quantity into symbolic entities *(5 represents any collection of five objects*) *and to *transform a number as a predicate in the native language to its conception as objects in the language of mathematics *(Wynn, 1996; 1998). For some children, particularly LD children, these transitions are not easy and need to be facilitated carefully using *appropriate language*, an *enabling questioning process*, and *efficient instructional models. *

**E. Mastering the Concept of Number**

**1. Lexical Entries and Egocentric Counting to the Cardinality of the Set**

Children develop lexical entries for number by hearing others count and copying this process. Number words are essential but not sufficient for fluent number conceptualization and usage.

Consider the number work of a five-year-old.

**Teacher:** How many cubes there are? (Points to the collection.)

❒ ❒ ❒ ❒ ❒ ❒ ❒

**Child:** (Counts by touching each cube) Seven.

**Teacher: **What number came **just** before seven?

**Child: **One? Three? Five? I don’t know.

**Teacher: **Can you give me six cubes?

**Child: **Do I have enough? Maybe I do. (She counts six cubes and gives them to the teacher.) **Teacher:** That is right.

Teacher rearranges them.

**Teacher:** How many cubes are there now?

**Child:** (She counts them) Seven.

**Teacher:** Yes!

Most children can count objects in a rote manner. For many of them, even at age 6, the cardinality of the set is the outcome of their counting process, not a property of the collection. Number is the product of ** egocentric **counting (“These are six blocks. I just counted them.”) rather than the property of the collection (“These are six cubes.”). This is a key step in number conceptualization. Consider number work with another Kindergartener.

❒ ❒ ❒ ❒ ❒ ❒ ❒

**Child:** (Counts them) Seven.

**Teacher: **Yes! You counted them from left to right (points to the direction). Do you think you will have the same number if you counted them from right to left? (Points to the direction.) **Child: **I do not know. Let me try.

**Child:** (Counts them) Seven.

**Teacher: **That is right.** **

**Child: (**Counts them again) Seven. It is always seven.

The child associates a number to the collection as the property of the collection. Scaffolding questions resulted in converting a child’s concrete experiences and egocentric counting into the cardinal number.

**2. Development of Visual Clusters**

For number conceptualization, child must transcend counting (Turner, 2003; Sophian & Kailihiwa, 1998). Young children spontaneously use the ability to recognize and discriminate small numbers of objects. This is called subitizing (Klein & Starkey, 1988, Clements, 1999). Subitizing is instantly seeing how many in a small collection of objects. But some young children cannot immediately name the number of objects in a collection. It is important for number conceptualizing. Work with dominos, dice, and playing cards helps in the process (Clements & Callahan, 1986). However, for efficient number concept, subitizing must be extended to numbers up to ten. We term that process as forming visual clusters in the mind. Visual Cluster Cards^{TM} (VCC), with modified arrangements of clusters, are especially effective for developing visual clustering and then number concept.

A VCC deck (60 cards) consists of 4 cards with 1 to 10 pips in 4 suites (heart, diamond, club, and spade); two cards with no pips represent zero; and 2 jokers (can be assigned any value). Numbers 3, 8, 9, and 10 have two representations. For example, number 3 is represented as:

The pips on higher number cards are organized so that the sub-clusters of smaller numbers can be instantaneously recognized. For example, on the 7-card, one can see clusters of 4 and 3; 5 and 2; and 6 and 1. No number names are displayed on the cards.

Creating images of visual clusters and developing the decomposition/recomposition process of numbers are at the heart of number conceptualization and arithmetic facts. Many children may achieve the decomposition/recomposition skill through counting; however, many children, particularly those with special needs, have difficulty achieving this with counting. Robinson et al. (2002) proposed that interventions for students with poor mastery of arithmetic combinations should include two aspects: (a) interventions to help build more rapid retrieval of information, and (b) concerted instruction in any areas of numbersense that are underdeveloped in a child. VC cards help achieve both.

The teacher introduces the VC cards for 1, 2 and 3. She identifies the cards by counting the pips. The card with two pips is identified as 2; 1 and 1; or two ones. Children learn that each visual cluster card is made of sub-clusters. For example, the teacher displays the card with three pips.

**Teacher:** Look at the card. How many diamonds are on the card?

**Child One:** Three.

**Teacher:** What three numbers make 3?

**Child Two**: 1 + 1 + 1.

**Teacher:** Look at the card with a circle around the diamonds. How many diamonds are circled?

**Teacher**: How many are not circled?

**Children:** One.

**Teacher:** What two numbers make 3?

**Child One**: 2 and 1.

**Child Two**: 1 and 2.

**Teacher:** Right! 2 + 1 makes 3 (traces the two circled diamonds and the one diamond); 1 + 2 also makes 3 (traces the one individual circled diamond and then the two diamonds); 1 + 1 + 1 also makes 3 (traces the three individual circled diamonds).

Once children have created the image of number 3 in the standard form, they do the same with the three objects organized in another form. For example:

The same process is used for developing the cluster images for higher numbers. For example, the teacher introduces the card representing number 5.

**Teacher:** How many diamonds are there?

Children count the diamonds on the card and say: Five.

**Teacher:** We will call this the 5-card. How many diamonds are there in the first column of the card? (She traces the first column.)

**Children**: 2.

**Teacher**: Yes. It represents the number 2.

**Teacher:** Now, how many diamonds are there in the last column? (She traces the last column.)

**Children:** 2.

**Teacher**: Good! It also represents number 2.

**Teacher:** Look at the middle column. How many diamonds are in the middle column?

**Children:** One!

**Teacher:** What if the middle diamond was not there, what number will the card represent?

**Children:** 4.

**Teacher:** Very good! (If a child is unable to answer, the teacher displays the card by covering the middle pip or show the 4-card again.) What if the first column was not there, what number will the rest of the card represent?

**Children**: 3.

**Teacher:** Very good! What if the last column was not there, what number will the rest of the card represent?

**Children:** 3.

The teacher continues till children have created the image of the cluster of number 5 in their minds. Every child should be able to identify the card in less than two seconds (without counting). They also know that the cluster of 5 has component sub-clusters of 2 and 3; 4 and 1; of 2, 2, and 1.

**Teacher**: Remember the card we have been looking at? I am going to show the card, but a portion of the card will be hidden. You need to tell me the missing number. (She hides the first column.) How many are visible?

**Children**: 3.

**Teacher:** How many are hidden?

**Children:** 2.

**Teacher:** What two numbers make 5?

**Children:** 2 and 3.

**Teacher:** Great! (She uncovers the hidden part of the card and shows the 5-card). Yes, 2 and 3 make 5.

Finally, each child has formed images of the number 5 as a visual cluster and its relationship with other numbers (decomposition/recomposition) as 4 + 1; 3 + 2; 2 + 2 + 1; and 1 + 1 + 1 + 1 + 1. Then she asks them to write these relationships.

Thus, knowing a number means an ability to write the number, use it as a count, recognize the visual cluster, and that it is made up of smaller numbers. This is true for all ten numbers:

- 2 = 1 + 1
- 3 = 2 + 1 = 1 + 2
- 4. = 3 + 1 = 1 + 3 = 2 + 2
- 5
**= 4 + 1 = 3 + 2 = 2 + 3 = 1 + 4** - 6
**= 5 + 1 = 4 + 2 = 3 + 3 = 2 + 4 = 1 + 5** - 7
**= 6 + 1 = 5 + 2 = 4 + 3 = 3 + 4 = 2 + 5 = 1 + 6** - 8
**= 7 + 1 = 6 + 2 = 5 + 3 = 4 + 4 = 3 + 5 = 2 + 6 = 1 + 7** - 9
**= 8 + 1 = 7 + 2 = 6 + 3 = 5 + 4 = 4 + 5 = 3 + 6 = 2 + 7 = 1 + 8** - 10 = 9 + 1 = 1 + 9 = 8 + 2 = 2 + 8 = 3 + 7 = 7 + 3 = 6 + 4 = 4 + 6 = 5 + 5

As the table above show, there are a total of 45 sight facts. Without the idealized image of these numbers, dight facts, and the decomposition/ recomposition process, children have difficulty in developing fluency in number relationships. Sight facts and decomposition/recomposition play the role in numberless and arithmetic as sight words and phonemic awareness plays in acquiring reading skills. Most dyscalculics and many underachievers in mathematics have not learned number concept properly.

Cuisenaire rods are another efficient tool for developing and extending the decomposition/ recomposition of numbers achieved through visual cluster cards. For example, the number 10 can be shown as the combination of two numbers as follows (the same process is used for other numbers):

Both Visual Cluster Cards and Cuisenaire rods help children to create and learn these decompositions.

**F. Concept of Addition**

Early mathematics interventions should focus on building fluency and proficiency with basic arithmetic facts as well as more accurate and efficient use of addition strategies (Gersten et al., 2005; Siegler, 1991; 1988). When children achieve fluency and efficiency in arithmetic combinations, teachers can assume that children are able to follow explanations of concepts or procedures.

Once children conceptualize idealized images of the ten numbers in the ‘mind’s eye’, they form sight facts and then easily learn addition facts. For example:

7 is made up of 4 and 3; 5 and 2; and 6 and 1.

Addition of numbers is facilitated through strategies of decomposition and recomposition of numbers. For example, to add 8 and 6 it is much easier to take two from 6 and give it to 8 so that the number combination can easily be seen as: 8 + 6 = 8 + (2 + 4) = (8 + 2) + 4 = 10 + 4 equals 14. Decomposition (the breaking of 6 as 2 + 4 and thinking of 10 as 8 + 2) and recomposition (thinking of 10 + 4 as 14) are key strategies for learning addition and subtraction facts.

When children do not automatize facts, they are unable to apply their knowledge to newer situations. To find the answer to a number problem, they digress from the main problem to generate the facts needed for solving problems. Because of the use of inefficient strategies, such as counting, their working memory space is filled in the process of constructing these facts, then it is not available to pay attention to instruction, observe patterns, or focus on concepts, nuances, relationships, and subtleties involved the concepts.

A child’s struggles with arithmetic facts in Kindergarten and first grade reflect a difficulty in transitioning from concrete to abstract number relationships and should trigger an intense intervention program in numberness and numbersense that focuses on visual clustering and decomposition/recomposition skills (Van Engen & Steffe, 1970). The use of tools such as Visual Cluster Cards and Cuisenaire rods can achieve that goal, prevent the development of acquired dyscalculia, and mitigate the effects of dyscalculia.

**References: **

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**Suggested Reading: **

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**CENTER** **FOR ** **TEACHING/LEARNING** **OF MATHEMATICS**

**LIST OF PUBLICATIONS 201**9

**Mathematics For All **

Programs and ServicesCT/LM has developed programs and materials to assist teachers, parents, therapists, and diagnosticians to help children and adults with their learning difficulties in mathematics. We conduct regular workshops, seminars, and lectures on topics such as: How children learn mathematics, why learning problems occur, diagnosis, and remediation of learning problems in mathematics. How does one learn mathematics? This workshop focuses on psychology and processes of learning mathematics—concepts, skills, and procedures. Participants study the role of factors such as: Cognitive development, language, mathematics learning personality, prerequisite skills, and conceptual models of learning mathematics. They learn to understand how key mathematics milestones such as number conceptualization, place value, fractions, integers, algebraic thinking, and spatial sense are achieved. They learn strategies to teach their students more effectively. 2. What are the nature and causes o problems in mathematics? This workshop focuses on understanding the nature and causes of learning problems in mathematics. We examine existing research on diagnosis, remedial and instructional techniques dealing with these problems. Participants become familiar with diagnostic and assessment instruments for learning problems in mathematics. They learn strategies for working more effectively with children and adults with learning problems in mathematics. 3. Content workshops. These workshops are for teachers and parents on teaching mathematics milestone concepts and procedures. For example, they address questions such as: How to teach arithmetic facts easily? How to teach fractions to students more effectively? How to develop the concepts of algebra easily? In these workshops, we use a new approach called Vertical Acceleration. In this approach, we begin with a simple concept from arithmetic and take it to the algebraic level. We offer individual diagnosis and tutoring services for children and adults to help them with their mathematics learning difficulties and learning problems, in general, and dyscalculia, in particular. We provide: Consultation with and training for parents and teachers to help their children cope withand overcome their anxieties and difficulties in learning mathematics. Consultation services to schools and individual classroom teachers to help themevaluate their mathematics programs and help design new programs or supplement existing ones in order to minimize the incidence of learning problems in mathematics. Assistance for the adult student who is returning to college and has anxiety about his/her mathematics. Assistance in test preparation (SSAT, SAT, GRE, MCAS, etc.)Extensive array of mathematics publications to help teachers and parents tounderstand how children learn mathematics, why learning problems occur and how to help them learn mathematics better. www.mathematicsforall.org |

Current PublicationsDyslexia and Mathematics Language Difficulties $15.00 Dyscalculia $15.00 Guide for an Effective Mathematics Lesson $15.00 Games and Their Uses: The Number War Game $15.00 How to Teach Arithmetic Facts Easily and Effectively $15.00 How to Teach Fractions Effectively $15.00 How to Teach Number to Young Children $15.00 How To Teach Subtraction Effectively $12.00 Literacy & Numeracy: Comprehension and Understanding $12.00 Math Education at Its Best: Potsdam Model $15.00 The Questioning Process: A Basis for Effective Teaching $12.00 Visual Cluster Cards (Playing Cards without Numbers) $15.00 eB ooks: The Questioning Process: A Basis for Effective Teaching. $10.00 How to Teach Number Effectively $15.00 DVDS How Children Learn: Numeracy $30.00 (One-hour long video interviewing Professor Mahesh Sharma on his ideas about how children learn mathematics) Teaching Place Value Effectively $30.00 Numeracy DVDs (Complete set of six for $150.00 and individual for $30.00)( Teaching arithmetic facts,Teaching place value,Teaching multiplication,Teaching fractions,Teaching decimals and percents, andProfessional development: teachers’ questions)Most children have difficulty in mathematics when they have not mastered the key mathematics milestones in mathematics. The key milestones for elementary grades are: Number conceptualization and arithmetic facts (addition and multiplication), place value, fractions and its correlates—decimal, percent, ratio and proportion. These videos and DVDs present strategies for teaching these key mathematics milestone concepts. They apply Prof. Sharma’s approach to teaching numeracy. These were videotaped in actual classrooms in the UK . Please add 20% of the total for postage and handling with your order: CENTER FOR TEACHING/LEARNING OF MATHEMATICS 754 Old Connecticut Path, Framingham, MA 01701 508 877 4089 (T), 508 788 3600 (F) mahesh@mathematicsforall.orgwww.mathematicsforall.org |

Articles in TMN address issues related to mathematics learning problems, diagnosis, remediation, and techniques for improving mathematics instruction. They translate research into practical and workable strategies geared towards the classroom teacher, parents and special needs teachers/tutors. Topics covered range from K through College mathematics instruction. **Selected Back Issues of The Math Notebook: **

- Children’s Understanding of the Concept of Proportion – Part 1 and 2 (double)
- A Topical Disease in Mathematics: Mathophobia (single)
- Pattern Recognition and Its Application to Math (double)
- Mathematics Problems of the Junior and Senior High School Students (double)
- Mathematically Gifted and Talented Students (double)
- Types of Math Anxiety (double)
- Memory and Mathematics Learning (double)
- Problems in Algebra – Part 1 and Part 2 (special)
- Reversal Problems in Mathematics and Their Remediation (double)
- How to Take a Child From Concrete to Abstract (double)
- Levels of Knowing Mathematics (double)
- Division: How to Teach It (double)
- Soroban: Instruction Through Concrete Learning (double)
- Mathematics Culture (double)
- Mathematics Learning Personality (double)
- Common Causes of Math Anxiety and Some Instructional Strategies (double)
- On Training Teachers and Teaching Math (double)
- Will the Newest “New Math” Get Johnny’s Scores Up? (double)
- Dyslexia, Dyscalculia and Some Remedial Perspectives For Mathematics Learning Problems (special)
- Place Value Concept: How Children Learn It and How To Teach It (special)
- Cuisenaire Rods and Mathematics Teaching (special)
- Authentic Assessment in Mathematics (special)

FOCUS has been an interdisciplinary journal. For the last thirty years, the objective of FOCUS was to make available the current research, methods of identification, diagnosis and remediation of learning problems in mathematics. It published original articles from fields of education, psychology, mathematics, and medicine having the potential for impact on classroom or clinical practice. Specifically, topics include reports of research on processes, techniques, tools and procedures useful for addressing problems in mathematics teaching and learning: descriptions of methodologies for conducting, and reporting and interpreting the results of various types of research, research-based discussions of promising techniques or novel programs; and scholarly works such as literature-reviews, philosophical statement or critiques. The publications in Focus have real contribution in the field of mathematics education, learning problems in mathematics and how to help children and adults in dealing with their mathematics difficulties.

**Back issues are available from 1979 to 2009 on request. **

About the Author is the founder and President of the Center for Teaching/Learning of Mathematics, Inc. of Framingham, Massachusetts and Berkshire Mathematics in England. Professor Mahesh SharmaBerkshire Mathematics facilitates his work in the UK and Europe. He is the former President and Professor of Mathematics Education at Cambridge College where he taught mathematics and mathematics education for more than thirty-five years to undergraduate and graduate students. He is internationally known for his groundbreaking work in mathematics learning problems and education, particularly dyscalculia and other specific learning disabilities in mathematics. He is an author, teacher and teacher-trainer, researcher, consultant to public and private schools, as well as a public lecturer. He was the Chief Editor and Publisher of Focus on Learning Problems in Mathematics, an international, interdisciplinary research mathematics journal with readership in more than 90 countries, and the Editor of The Math Notebook, a practical source of information for parents and teachers devoted to improving teaching and learning for all children. He provides direct services of evaluation and tutoring for children as well as adults who have learning disabilities such as dyscalculia or face difficulties in learning mathematics. Professor Sharma works with teachers and school administrators to design strategies to improve mathematics curriculum and instruction for all. Contact Information: Mahesh C. Sharma mahesh@mathematicsforall.org508 494 4608 (C) 508 788 3600 (F) Blog: www.mathlanguage.wordpress.comCenter for Teaching/Learning of Mathematics 754 Old Connecticut Path Framingham, MA 01701 mahesh@mathematicsforall.org http://www.mathematicsforall.org |

Mathematics Education Professional Development Workshop SeriesFramingham State UniversitywithProfessor Mahesh SharmaAcademic Year 2019-2020Several national professional groups, the National Mathematics Advisory Panel and the Institute for Educational Sciences in particular, have concluded that all students can learn mathematics and most can succeed through Algebra 2. However, the abstractness and complexity of algebraic concepts and missing precursor skills and understandings–number conceptualization, arithmetic facts, place value, fractions, and integers–may be overwhelming to many students and teachers. Being proficient at arithmetic is certainly a great asset when we reach algebra; however, how we achieve that proficiency can also matter a great deal. The criteria for mastery, Common Core State Standards in Mathematics (CCSSM), set for arithmetic for early elementary grades are specific: students should have (a) understanding (efficient and effective strategies), (b) fluency, and (c) applicability and will ensure that students form strong, secure, and developmentally appropriate foundations for the algebra that students learn later. The development of those foundations is assured if we implement the Standards of Mathematics Practices (SMP) along with the CCSSM content standards. In these workshops, we provide strategies; understanding and pedagogy that can help teachers achieve these goals. All workshops are held on the Framingham State University campus from 8:30am to 3:00pm. Cost is $49.00 per workshop and includes breakfast, lunch, and materials. PDP’s are available through the Massachusetts Department of Elementary and Secondary Education for participants who complete a minimum of two workshops together with a two page reflection paper on cognitive development. A. Creating A Dyscalculia Friendly ClassroomLearning Problems in Mathematics (including math anxiety)For special education, regular education teachers, interventionists, and administrators October 11, 2019 In this workshop, participants will learn (a) why learning problems in mathematics (e.g., dyscalculia, etc.) occur, (b) how children learn mathematics, (c) what are effective methods of teaching mathematics, and (d) how to fill gaps in mathematics learning. The major aim is to deliver mathematics instruction that prevents learning problems in mathematics from debilitating a student’s learning processes in mathematics. B. Number Concept, Numbersense, and Numeracy SeriesAdditive Reasoning (Part I): How to Teach Number Concept EffectivelyFor K through grade second grade teachers, special educators and interventionistsNovember 1, 2019Number concept is the foundation of arithmetic. Ninety-percent of students who have difficulty in arithmetic have not conceptualized number concept. In this workshop we help participants learn how to teach number concept effectively. This includes number decomposition/recomposition, visual clustering, and a new innovative concept called “sight facts.” Additive Reasoning (Part II): How to Teach Addition and Subtraction EffectivelyFor K through grade third grade teachers, special educators and interventionistsNovember 22, 2019According to Common Core State Standards in Mathematics (CCSS-M), by the end of second grade, children should master the concept of Additive Reasoning (the language, concepts and procedures of addition and subtraction). The mastery means (a) understanding, fluency, and applicability. In this workshop, the participants will learn effective, efficient, and elegant ways of achieving this with their students. Multiplicative Reasoning (Part III): How to Teach Multiplication and Division EffectivelyFor K through four second grade teachers, special educators and interventionistsDecember 13, 2019According to CCSS-M, by the end of fourth grade, children should master the concept of Multiplicative Reasoning (the language, concepts and procedures of multiplication and division). The mastery means (a) understanding, fluency, and applicability. In this workshop, the participants will learn effective, efficient, and elegant ways of achieving this with their students. C. Proportional Reasoning SeriesHow to Teach Fractions Effectively (Part I): Concept and Multiplication and DivisionJanuary 24, 2020 For grade 3 through grade 9 teachers and special educatorsAccording to CCSS-M, by the end of sixth grade, children should master the concept of Proportional Reasoning (the language, concepts and procedures ratio and proportion). The concepts of ratio and proportion are dependent on the mastery of the concept of fractions. The mastery means (a) understanding, fluency, and applicability of fractions and operations on them. In this workshop, the participants will learn effective, efficient, and elegant ways of achieving the concept of fractions and multiplication and division of fractions and help their students achieve that. How to Teach Fractions Effectively (Part II): Concept and Addition and SubtractionFor grade 3 through grade 9 teachersFebruary 28, 2020According to CCSS-M, by the end of sixth grade, students should master the concept of Proportional Reasoning (the language, concepts and procedures ratio and proportion). The concepts of ratio and proportion are dependent on the mastery of the concept of fractions. The mastery means (a) understanding, fluency, and applicability of fractions and operations on them. In this workshop, the participants will learn effective, efficient, and elegant ways of achieving the concept of fractions and operations on fractions-from simple fractions to decimals, rational fractions and help their students achieve that. D. AlgebraArithmetic to Algebra: How to Develop Algebraic ThinkingFor grade 4 through grade 9 teachersMarch 20, 2020According to CCSS-M, by the end of eighth-grade, students should acquire algebraic thinking. Algebra is a gateway to higher mathematics and STEM fields. Algebra acts as a glass ceiling for many children. From one perspective, algebra is generalized arithmetic. Participants will learn how to extend arithmetic concepts to algebraic concepts and procedures effectively and efficiently. On the other perspective, algebraic thinking is unique and abstract and to achieve this thinking students need to engage in cognitive skills that are uniquely needed for algebraic thinking. In this workshop we look at algebra from both perspectives: (a) Generalizing arithmetic thinking and (b) developing cognitive and mathematical skills to achieve algebraic thinking. E. General TopicsMathematics as a Second Language: Role of Language in Conceptualization and in Problem SolvingFor K through grade 12 teachersApril 3, 2020Mathematics is a bona-fide second language for most students. For some, it is a third or fourth language. It has its own vocabulary, syntax and rules of translation from native language to math and from math to native language. Some children have difficulty in mathematics because of language difficulties. Most children have difficulty with word problems. In this workshop, the participants will learn how to teach effectively and efficiently this language and help students become proficient in problem solving, particularly, word problems. Learning Problems in Mathematics (including dyscalculia)For special education and regular education teachers May 15, 2020In this workshop, participants will learn (a) why learning problems in mathematics (e.g., dyscalculia, etc.) occur, (b) how children learn mathematics, (c) what are effective methods of teaching mathematics, and (d) how to fill gaps in mathematics learning. Standards of Mathematics Practice: Implementing Common Core State Standards in MathematicsFor K through grade 11 teachers (regular and special educators)June 12, 2020CCSS-M advocates curriculum standards in mathematics from K through Algebra II. However, to achieve these standards, teachers need to change their mind-sets about nature of mathematics content; every mathematics idea has its linguistic, conceptual and procedural components. Most importantly, these standards cannot be achieved without change in pedagogy-language used, questions asked and models used by teachers to understand and teach mathematics ideas. Therefore, framers of CCSS-M have suggested eight Standards of Mathematics Practice (SMP). In this workshop, we take examples from K through high school to demonstrate these instructional standards with specific examples from CCSS-M content standards. For registration, PDPs, Parking, and other information, please Contact: Anne Miller: 508 620 1220Continuing Education Department Framingham State University Framingham, MA 01701 |

Many children have difficulty learning and automatizing arithmetic facts (addition, subtraction, multiplication and division). In this video, I demonstrate the strategies and processes of mastering number concept. I believe almost all children can automatize arithmetic facts with strategies if they have mastred number concept properly.

]]>Mathematics Education Workshop Series at Framingham State UniversitywithProfessor Mahesh SharmaAcademic Year 2019-2020Several national professional groups, the National Mathematics Advisory Panel and the Institute for Educational Sciences in particular, have concluded that all students can learn mathematics and most can succeed through Algebra 2. However, the abstractness and complexity of algebraic concepts and missing precursor skills and understandings–number conceptualization, arithmetic facts, place value, fractions, and integers–may be overwhelming to many students and teachers. Being proficient at arithmetic is certainly a great asset when we reach algebra; however, how we achieve that proficiency can also matter a great deal. The criteria for mastery, Common Core State Standards in Mathematics (CCSSM), set for arithmetic for early elementary grades are specific: students should have (a) understanding (efficient and effective strategies), (b) fluency, and (c) applicability and will ensure that students form strong, secure, and developmentally appropriate foundations for the algebra that students learn later. The development of those foundations is assured if we implement the Standards of Mathematics Practices (SMP) along with the CCSSM content standards. All workshops are held on the In these workshops, we provide strategies: understanding and pedagogy, that can help teachers achieve these goals. Framingham State University campus from 8:30am to 3:00pm. A. Creating A Dyscalculia Friendly Classroom: Learning Problems in Mathematics (including math anxiety)For special education, classroom teachers, interventionists, parents, and administratorsOctober 11, 2019 In this workshop, participants will learn (a) why learning problems in mathematics (e.g., dyscalculia, etc.) occur, (b) how children learn mathematics, (c) what are effective methods of teaching mathematics, and (d) how to fill gaps in mathematics learning. The major aim is to deliver mathematics instruction that prevents learning problems in mathematics from debilitating a student’s learning processes in mathematics. B. Number Concept, Numbersense, and Numeracy SeriesAdditive Reasoning (Part I): How to Teach Number Concept Effectively For K through grade second grade teachers, special educators and interventionists November 1, 2019 Number concept is the foundation of arithmetic. Ninety-percent of students who have difficulty in arithmetic have not conceptualized number concept. In this workshop we help participants learn how to teach number concept effectively. This includes number decomposition/recomposition, visual clustering, and a new innovative concept called “sight facts.” Additive Reasoning (Part II): How to Teach Addition and Subtraction Effectively For K through grade third grade teachers, special educators and interventionists. November 22, 2019 According to Common Core State Standards in Mathematics (CCSS-M), by the end of second grade, children should master the concept of Additive Reasoning (the language, concepts and procedures of addition and subtraction). The mastery means (a) understanding, fluency, and applicability. In this workshop, the participants will learn effective, efficient, and elegant ways of achieving this with their students. Multiplicative Reasoning: How to Teach Multiplication and Division Effectively. For K through four second grade teachers, special educators and interventionists December 13, 2019 According to CCSS-M, by the end of fourth grade, children should master the concept of Multiplicative Reasoning (the language, concepts and procedures of multiplication and division). The mastery means (a) understanding, fluency, and applicability. In this workshop, the participants will learn effective, efficient, and elegant ways of achieving this with their students. C. Proportional Reasoning SeriesHow to Teach Fractions Effectively (Part I): Concept and Multiplication and Division January 24, 2020 For grade 3 through grade 9 teachers and special educatorsAccording to CCSS-M, by the end of sixth grade, children should master the concept of Proportional Reasoning (the language, concepts and procedures ratio and proportion). The concepts of ratio and proportion are dependent on the mastery of the concept of fractions. The mastery means (a) understanding, fluency, and applicability of fractions and operations on them. In this workshop, the participants will learn effective, efficient, and elegant ways of achieving the concept of fractions and multiplication and division of fractions and help their students achieve that. How to Teach Fractions Effectively (Part II): Concept and Addition and SubtractionFor grade 3 through grade 9 teachersFebruary 28, 2020According to CCSS-M, by the end of sixth grade, students should master the concept of Proportional Reasoning (the language, concepts and procedures ratio and proportion). The concepts of ratio and proportion are dependent on the mastery of the concept of fractions. The mastery means (a) understanding, fluency, and applicability of fractions and operations on them. In this workshop, the participants will learn effective, efficient, and elegant ways of achieving the concept of fractions and operations on fractions-from simple fractions to decimals, rational fractions and help their students achieve that. D. Algebraic Thinking and Reasoning Arithmetic to Algebra: How to Develop Algebraic ThinkingFor grade 4 through grade 9 teachersMarch 20, 2020According to CCSS-M, by the end of eighth-grade, students should acquire algebraic thinking. Algebra is a gateway to higher mathematics and STEM fields. Algebra acts as a glass ceiling for many children. From one perspective, algebra is generalized arithmetic. Participants will learn how to extend arithmetic concepts to algebraic concepts and procedures effectively and efficiently. On the other perspective, algebraic thinking is unique and abstract and to achieve this thinking students need to engage in cognitive skills that are uniquely needed for algebraic thinking. In this workshop we look at algebra from both perspectives: (a) Generalizing arithmetic thinking and (b) developing cognitive and mathematical skills to achieve algebraic thinking. E. General TopicsMathematics as a Second Language: Role of Language in Conceptualization and in Problem SolvingFor K through grade 12 teachersApril 3, 2020Mathematics is a bona-fide second language for most students. For some, it is a third or fourth language. It has its own vocabulary, syntax and rules of translation from native language to math and from math to native language. Some children have difficulty in mathematics because of language difficulties. Most children have difficulty with word problems. In this workshop, the participants will learn how to teach effectively and efficiently this language and help students become proficient in problem solving, particularly, word problems. Learning Problems in Mathematics (including dyscalculia)For special education, classroom teachers, parents, and administrators May 15, 2020In this workshop, participants will learn (a) why learning problems in mathematics (e.g., dyscalculia, etc.) occur, (b) how children learn mathematics, (c) what are effective methods of teaching mathematics, and (d) how to fill gaps in mathematics learning. Standards of Mathematics Practice: Implementing Common Core State Standards in MathematicsFor K through grade 11 teachers (regular, special educators, and administrators)June 12, 2020CCSS-M advocates curriculum standards in mathematics from K through Algebra II. However, to achieve these standards, teachers need to change their mind-sets about nature of mathematics content; every mathematics idea has its linguistic, conceptual and procedural components. Most importantly, these standards cannot be achieved without change in pedagogy-language used, questions asked and models used by teachers to understand and teach mathematics ideas. Therefore, framers of CCSS-M have suggested eight Standards of Mathematics Practice (SMP). In this workshop, we take examples from K through high school to demonstrate these instructional standards with specific examples from CCSS-M content standards. |

Here is the link for more information and registration: https://reg.abcsignup.com/view/cal4a.aspx?ek=&ref=&aa=&sid1=&sid2=&as=5&wp=18&tz=&ms=&nav=&cc=&cat1=&cat2=&cat3=&aid=FSU&rf REGISTER!!! |

These workshops are well subscribed. Call early to hold your place. Participants in each workshop are provided with lecture notes that inlcude the latest research on the topic and more than fifity years’ of Professor Sharma’s expereince. |

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On the other hand, mathematics educators have focused on exploring different ways of teaching mathematics to students with and without mathematics learning disabilities. They have wondered whether the learning resources and experiences provided to them, in the classroom or in intervention settings, are appropriate to their needs.

As a result, if we want to reach these students, we need to examine the interaction of cognitive skills, instructional strategies, and instructional materials in designing diagnosis, instruction (initial and remedial), and assessments. In other words, we need to examine what instruction suites the needs of a student with particular deficits and strengths. To answer some of the questions related to this, research in the field of dyslexia might be instructive.

**A. Processing and Mathematics Learning **

Research in the field of dyslexia has shown that typically developing children and children with dyslexia differ in all temporal processing (TP) skills. It is also further shown that cross-modal TP also contributes independently to character recognition (i.e., mathematics symbols and characters in ideographic languages such as Chinese, Hindi, Japanese, etc.) in children if the significant effects of phonological awareness, orthographic knowledge, and rapid automatized naming (RAN) are also considered. In multiple situations, it is shown that visual and cross-modal TP skills contribute to mathematics learning in addition to content related skills such as number concept and numbersense. In other words, the poor TP skills have higher impact when numerical skills are poor to start with. That means poor TP skills exacerbate the poor number skills and the difficulty in learning them. For example, visual and cross-modal TP skills have direct effects on character and symbol recognition and reading in the group with deficits such as dyslexia and/or dyscalculia, but, somehow they do not have impact on those who have better content skills in these areas.

Research findings also suggest that TP is more important for reading in children with dyslexia or dyscalculia than in typically developing children, and that TP plays an important role in dyslexia and dyscalculia. One of the ways TP plays role in mathematics learning is through its affect on working memory. In several postings, on working memory, on this blog, I have discussed the various roles working memory plays in learning mathematics and its role when there is deficit in it.

**B. Types of Working memory and their role in mathematics disabilities**

Developmental learning disabilities such as dyslexia and dyscalculia have a high rate of co-occurrence in early elementary populations, suggesting that they share underlying cognitive and neurophysiological mechanisms and processes. Multiple cognitive and neuropsychological skills, such as executive function and processing skills have been implicated in the incidence of dyslexia and dyscalculia. For example, dyslexia and other developmental disorders with a strong heritable component have been associated with reduced sensitivity to coherent motion stimuli—an important component of visuo-spatial processing and an index of visual temporal processing. It affects focus, inhibition control, visual processing skills, and spatial orientation/space organization—critical components for mathematics learning. Children with mathematics skills in the lowest 10% in their cohort are less sensitive than age-matched controls to coherent motion, but research shows that they have statistically equivalent thresholds to controls on a coherent form control measure. Research has also shown that deficits in sensitivity to visual motion are evident in children who have poor mathematics skills compared to other children of the same age.

Children with mathematics difficulties tend to present similar patterns of visual processing deficit as other developmental disorders suggesting that reduced sensitivity to temporally defined stimuli such as coherent motion represents a common processing deficit apparent across a range of commonly co-occurring developmental disorders, including mathematics disabibities.

However, the term mathematics difficulties and their behavioral skills markers across studies, across curricula, and across interventions are quite diverse. This is so because mathematics is so diverse in nature, content, concepts, and thinking that comparing mathematics difficulties with other learning disabilities is artificial. Mathematics learning ranges from mastery of *numerical reasoning*—number concept, numbersense, and numeracy (study of patterns in number—arithmetic), *spatial reasoning*—spatial sense and various geometrical objects and thinking (study of patterns in shapes and their relationships in diverse situations), *generalization of arithmetic and sense of variability*(study of patterns in variability—algebra thinking), *integration of algebraic and geometrical thinking and modeling*(coordinate geometry, trigonometry—for every geometrical figure there is an equation or system of equations and for every equation there is a geometrical representation), *proportional reasoning*(study of patterns in the rate of change—calculus), etc. All of these call for a complex of cognitive, neuropsychological, spatial, logical and linguistic skills at variety of levels. Any deficit in one or more of these skills could be the cause of mathematics difficulties in a various disciplines/fields of mathematics.

Although developmental and classification models in different fields of mathematics have been developed (for example, Geary and Hoard, 2005; Desoete, 2007; von Aster and Shalev, 2007), to our knowledge, no single framework or model can be used for a comprehensive and fine interpretation of students’ mathematical difficulties across different disciplines of mathematics. This is true not only for scientific purposes—diagnosis, behavioral markers, research, prevention, but also for informing mathematics and special educators, and for designing appropriate instruction—appropriate to the content, appropriate to the learner, and appropriate to the condition of his/her mathematics disability.

As an educator^{1}, I believe that reaching a model that focuses on a definite and important aspect of mathematics that combines, elaborates and enhances on existing hypotheses on MLD, based on known cognitive processes and mechanisms, could be used to provide a *mathematical profile* of a student. Since, not all students reach calculus, just like not all students become professional writers, in such a situation, it is important to focus on the fundamental components of mathematics, such as: numeracy as a focus of study to define the parameters of MLD. Therefore, in this discussion, we will limit MLD to disabilities in learning numeracy: number concept, numbersense, place value, numerical procedures.

Most commonly, MLD has been linked to problems with working memory, i.e. the brain’s ability to hold and manipulate information over a short period of time. Students with poor working memory cannot hold and manipulate information (oral, visual, tactile, or combination of them) in the mind’s eye (working memory space).

Working memory was initially thought to be domain-general, meaning that its importance is the same regardless of whether the content is related to mathematics, reading, or some other subject. However, some studies have shown that working memory may be domain specific—different subtypes of working memory operate for different types of tasks. Research has also shown that there are more than just one cause for MLD by showing that the MLD participants could be divided into at least two groups in which each had a different type of working memory deficit.

One group with MLD shows poor reading skills and scores poorly on verbal working memory tasks (tasks involving holding in memory and manipulating verbal information). The other group has purer deficits in mathematics and scores poorly on visuo-spatial working memory tasks (tasks involving holding and manipulating spatial and visual information). This seems to indicate that there could be at least two different causes of MLD, both of which are subtypes of working memory.

The opposite of this phenomenon is also true: There is a continuum of mathematics learning personalities ranging from *quantitative mathematics learning personalities*(with strength in sequential processing, procedures, etc.) to*qualitative mathematics learning personality*(with strength in visuo-spatial processing and pattern recognition, etc.). As a result, quantitative mathematics learning personality students perform better in arithmetic procedures and algebra and have difficulty in conceptual aspects of mathematics, geometry and problem solving (e.g., properties of numbers, word problems, proofs in geometry and algebra, etc.). Qualitative mathematics learning personality students, on the other hand, do better on conceptual aspects of mathematics, geometry, and problem solving and find it difficult to execute multi-step procedures in arithmetic (e.g., long-division, solving systems of equations, etc.). (Sharma, 2010)

Mathematics requires an extensive network of brain activities and that a problem with any one of the two types of memories could lead to MLD. Since verbal processing seems to be required for the brain to conduct mathematics, underdeveloped verbal processing could lead to MLD. In the same way, the brain also to use visuo-spatial processing for mathematics, so a deficit in this area could also lead to MLD. Let us look at the group that performs poorly on both verbal and visuo-spatial working memory. In such a situation, we can see problems in earlier mathematics (numeracy) as well as early reading in lower grades.

Whilst the symptoms of MLD can look similar, the problem may arise from different sources. Since mathematics is such a rigorous (requiring a lot of mental resources) and exacting (requiring one perfect answer in many cases) discipline, it could reveal brain deficits that would not show up with other disciplines. This could be why participants who score poorly on visuo-spatial working memory tasks can do well in other subjects besides mathematics. The fact that MLD can arise because of multiple distinct deficits may be of great importance both to mathematics teachers and researchers investigating causes of MLD in children.

It is, therefore, natural to raise the question: Is it possible to use instructional materials and mathematics specific pedagogy that can enhance not only the mathematics content, but also improve the working memory and processing functions, thereby improving students’ learnability? To find the answer, let us look at a very specific concept and various ways children arrive at the answer as an outcome of instruction and their understanding of the nature of the concept, use of instructional material, language, and expectations from them. Children’s responses to this problem show, in many cases, the weaknesses not because of their own working memory, but imposed by and outcome of inefficient instruction, poor instructional materials, and lower expectations. Such difficulties can be removed with proper instruction, effective, efficient, and elegant instructional materials, rich language, and higher expectations.

For example, let us examine the approaches children are exposed to a range of instructional materials that are used in classroom and the strategies that are developed as a result of this instruction, from pre-Kindergarten to third grade. We want to examine, whether some of them have the potential to improve students’ learning not only the mathematics content and processes, but also the learnability (ability to learn—cognitive skills, for example improving working memory, visual processing, generalizing, etc.). To find the answer to this question, let us consider a simple addition problem and the related strategies.

**C. Problem:***Find the sum 9 + 7.*

To find this sum, there are many options, approaches and strategies, a child may use to arrive at the answer. However, an average or below average child has only those choices that are possible by the instructional material used, strategies possible using those materials, the approach the teacher introduced, or helped children develop in the classroom. The following descriptions are of approaches, I have seen children use in different schools, different settings, and different countries. We want to examine them from the perspective of effectiveness, efficiency, and elegance.

For example: A child:

(**a)**Counts 9 items and, then 7 items from a collection of random objects and gets the answer. Our early childhood classrooms abound with these kinds of materials—beans, shells, pennies, gummy bears, etc.

(**b**) Counts 9 and 7 cubes (Unifix cubes, Centi-cubes, inter-locking cubes, wooden cubes, BaseTen cubes, etc.) from the cubes collection, in the classroom.

(**c**) Counts 7 fingers and then 9 fingers to find the answer.

(**d)** Decides that 9 is the bigger number, he counts 7 numbers after 9 and finds the answer.

(**e)** Counts 9 beads on one TenFrame and 7 on another TenFrame and then counts them all, sequentially from the first TenFrame and then the second TenFrame.

**(f)** Recognizes that 9 is the bigger number, he makes the number 7 configuration by touch points on the paper or on his body part and counts them after 9 to find the answer.

(**g**) Counts 7 objects (cubes, fingers, marks, etc.) after the number 9 and reaches 16, and declares the answer.

(**h**) Places 9 unit-rods and 7 unit-rods end-to-end, from the BaseTen set to make a train and then replace the 10 unit-rods by the 10-rod and then place the 10-rod and 6-unit rods end-to-end to make a train and the child counts 10-11-12-13-14-15-16 as he touches each unit-cube, and arrives at the answer.

(**i**) Locates 9 and 7 on two Ten frames (5 and 4 on one) and (5 and 2 on the second one) and then adds 5 and 5 and 4 and 2, and declares the answer as 16;

(**j)** Places a 9-rod and 7-rod, from the Cuisenaire rods collection, end-to-end to make a train and then experiments which rod along with the 10-rod is equal to the 9-rod and 7-rod train. The child finds that the train made with the10-rod and 6-rod placed end-to-end is of the same length as the 9-rod plus 7-rod train. The child finds the answer.

(**k**) Places a 9-rod and 7-rod, from Cuisenaire collection, end-to-end to make a train and realizes that the train is longer than the 10-rod. He places the 10-rod parallel to the 9-rod + 7-rod train. He observes that the 10-rod is 9 + 1 and realizes that there is 6 left from the 7 rod. Now he adds 10 and 6 to get 16.

(**l**) Places a weight of 1-unit at the peg at 9 and another 1-unit weight at the peg at 7 on the left arm of the Invicta Balance and, then a weight of 1 unit at the peg at 10 and experiments with placing weights on other pegs in order to balance the arm. In experimenting in placing unit weights at the peg at 6, the arm balances. Since, the arm is balanced (is horizontal), the child realizes that 9 + 7 = 10 + 6 = 16.

(**m**) Makes 9 and 7 marks (⁄, ×, ✓, , etc.) with pencil-on-paper and counts them to find the answer.

(**n**) Locates a number line (on the wall, on the desk, or in the book) and counts numbers, first the 9 and then 7 more on the number line and finds the answer.

(**o**) Locates a number line (on the wall, on the desk, or in the book) and first locates number 7 on the number-line, and then counts 9 more numbers after 7 and ends on 16 and declares the answer as 16;

(**p**) Locates a number line (on the wall, on the desk, or in the book), then locates the number 9 and then counts 7 numbers after 9, and ends at 16 and declares the answer;

(**q**) Locates 9 beads on a TenFrame, and then realizes he can count 1 more on the first TenFrame and counts 6 on the second Ten frame. He says: 9 + 1 is 10 and 6 more is 16.

(**r**) Selects a 9-card and 7-card from the Visual Cluster cards collection and visually move the one pip from the 7-card to 9-card to make it a 10-card and then visually realizes that the 7-card has become a 6-card, so the sum 9 + 7 becomes10 + 6. Declares the answer as 16.

(**s**) Locates 9 and 7 on two Paper form of the Ten frames (5 and 4 on one and 5 and 2 on the second one) and then adds 5 and 5 and 4 and 2, and declares the answer as 16;

(**t)** Draws an Empty Number Line (ENL) and locates 9 on the left of side of the line and then takes a jump of 1 from 9 and lands at 10 and then takes a jump of 6 from 10 and then lands at 16. Declares the answer as 16.

(**u**) Places a 9-rod and 7-rod, from Cuisenaire collection, end-to-end to make a train and realizes that the train is longer than the 10-rod. Places the 10-rod parallel to the 9-rod + 7-rod. He observes that the 10-rod is 9 + 1 and realizes that there is 6 left from the 7 rod. Then, combines 10 and 6 to get the answer. At this point, he transfers the learning from this model to visualization and then abstract form: 9 + 7 = 9 + 1 + 6 = 10 + 6 = 16.

(**v**) Pictures, in his mind, a 9-card and 7-card from the Visual Cluster cards collection and visually moves the one pip from the 7-card to 9-card to make it a 10-card and then visually realizes that the 7-card has become a 6-card, so the sum 9 + 7 becomes10 + 6. Declares the answer as 16.

(**w**) Pictures, in his mind, a 9-rod and 7-rod, from Cuisenaire collection, thinks of them together, visually takes 1 from the 7-rod and gives it to 9 and then realizes he has 10 + 6. Then, combines 10 and 6 to get the answer. At this point, he has transferred the learning from the concrete model to visualization of 9 + 7 = 9 + 1 + 6 = 10 + 6 = 16. Teacher’s questions and scaffolding makes it possible.

(**x)** Draws and Empty Number Line in his mind and locates 9 on the line and then starts with 9 and takes an imaginary jump of 1 and lands at 10. He knows that the 1 came from 7 so he takes another imaginary jump of 6 from 10 and lands at 16. Declares the answer is 16. With few practices, he generalizes the strategy into: 9 + 7 = 9 + 1 + 6 = 10 + 6 = 16.

(**y**) Begins with the problem 9 + 7 = ?, Using the decomposition/ recomposition and commutative property of addition, and decomposition/ recomposition strategies converts the problem into any of the following convenient forms and finds the sum:

(i) 9 + 7 = 9 + 1 + 6 = 10 + 6 = 16;

(ii) 9 + 7 = 6 + 3 + 7 = 6 + 10 = 16;

(iii) 9 + 7 = 2 + 7 + 7 = 2 + 14 = 16;

(iv) 9 + 7 = 9 + 9 – 2 = 18 – 2 = 16;

(v) 9 + 7 = 7 + 9 = 7 + 10 – 1 = 17 – 1 = 16; or,

(vi) 9 + 7 = 5 + 4 + 2 + 5 = 5 + 6 + 5 = 5 + 5 + 6 = 16.

**D. Analysis of Strategies** **Used by Children**:

**1. Counting based approaches**

The approaches from (**a)**through (**i**) involve the instructional materials (concrete in nature) that promote the strategy of counting objects. This results in a child conceptualizing addition as the result of counting. The only strategy these approaches develop is counting. The counting process does not develop relationships and patterns between numbers. It does not leave any residue of number relationships on the memory. The activity, thus, does not build up any working memory. Moreover, consistent use of these materials makes the child dependent on concrete materials and, then remains functioning at the *concrete level of knowing*.

These children, generally, have great deal of difficulty in automatizing addition facts, if at all they reach that level. Many of them remain at the concrete level, even at the high school level. And, when these children are referred to intervention, special education classes, or one-to-one support, they may get the same approach (i.e., addition is counting up). They do not make much progress. Their achievement in mathematics continues to be deficient.

The counting strategy helps them derive the sum, but the mathematics content such derived is very limited, mathematics strategies are inefficient, they cannot be generalized, and the resultant cognitive learning is very limited. These materials and the counting strategies derived from this process are ineffective and inefficient and the resultant arithmetic methods of teaching and learning are limited. Since, there is no improvement in working memory space, the inadequacy in learning continues.

*2. Length-based approaches*

The approaches from **(j)**and (**k**) are also concrete, but the strategy derived is not counting based. *Here the number magnitude is being associated with length of the rods.*Children begin to see numbers as groups rather than one-to-one. Just like a word is a group of letter and has its own entity, similarly, when we consider number as the representation of a particular length and combining two or more numbers into a new numbers, they get the same feel as making a word from different letters.

This transition from one-to-one counting to forming a group is a ‘*cognitive jump’*for a child. As we will see, the rods model has the potential to be converted into effective and efficient strategies, not only for addition, but also for other arithmetic facts. Because of the color and size (lengths) of the Cuisenaire rods, one can visualize the numbers (by remembering the rods) and can see the equation in the mind’s eye and, therefore, remember the equation easily. This enhances working memory, visual memory, visual-perception, and pattern recognition. They can transit from addition to subtraction easily.

*3. Weight-based approach*

The approach in *( l)*is based on the weight aspect of number. It is not a very transparent strategy. Answer is found easily, but it does not result in a strategy. However, it is the best approach to develop the concept of

**4. Approaches based on pictorial representation **

The approaches from (**m**) through (**q**) are also based on counting of objects. The strategy used is still seen as conceptualizing addition as *“counting up.”*Children are using pictorial representation of objects (*pictorial/ representational level of knowing*), thereby, are functioning at a higher level than the concrete level. However, conceptually, it is still counting. Once again, generally, they do not automatize addition facts and their achievement in mathematics continues to be deficient. This strategy helps them derive the sum, but neither leaves any residue in the memory nor it develops and increases the working memory space. These materials and strategy derived from this process are still inefficient. This does not help in improving a child’s cognition.

**5. Approaches based on decomposition/recomposition **

The approaches from (**r**) through (**w**) whether derived using Ten Frames, Visual Cluster cards, Cuisenaire rods, or Empty Number Line are helping children to reach *abstract level knowing*easier and quicker. These instructional materials are helpful in developing both kinds of working memory skills: verbal working memory and visuo-spatial working memory. The strategies are effective, efficient, and elegant. These children will be able to master arithmetic facts easily. These strategies help children improve their working memory, cognition and ability to generalize their learning. These strategies become more effective when they can see the relationship across instructional materials, as they help integrating the two kinds of working memories.

Concrete models, such as: **TenFrames**, **Visual Cluster cards**, **Cuisenaire rods**, with the help of (a) the *teacher’s scaffolded questions;* (b) *sight facts*, (c) *decomposition/recompostion*; (d) *making ten*; and (e) *knowing teens’ *numbers develop strategy and the script for finding the addition fact. For example, the following is the script and the strategy for (**i**) in approach (**y**). “I want to find: 9 + 7 = ?. In 9 + 7, 9 is the bigger number. So, I want to make it 10 by adding 1 to it ( 9 + 1 = 10). Number 1 comes from 7, (7 = 1 + 6) . Now I have 10 and 6. I know 10 + 6 is 16. So, 9 + 7 = 9 + 1 + 6 = 10 + 6 = 16. Therefore, 9 + 7 = 16. And, by commutative property of addition, 9 + 7 = 7 + 9, so 7 + 9 = 16.” Similar scripts can be developed and used for other strategies from (**ii**) to (**vi**) in approaches in (**y**).

Teacher’s questions, consistent practice of visualization of these strategies, with constructive feedback to performance, by reminding children of the use of efficient strategies, and use of scripts that empahsize these strategies help children in masteringof addition facts. **Mastery** here refers to: *accuracy*, *understanding*, *fluency*, and *applicability*. When addition facts are mastered all other arithmetic facts are eaasy to master.

**6. Behavioral markers of mastery (CPVA)**

The approaches in section (**x**) transcend counting of individual numbers and demonstrate the integration of the key prerequisite skills necessary for mastery of addition facts: (i) understanding and using *decomposition/ recomposition of number,*(ii) mastery of 45 *sight facts*, (iii) *making ten*, (iv) *making teens’ numbers, (v) visualization *(working memory space), and (vi) the *concept of equality*. In the hands of an effective teachers, the instructional materials: Ten Frames, Visual Cluster cards, Cuisenaire rods, and Empty Number Line (ENL), in this order, when assisted by her appropriate questions and proper language become ideal materials for helping children to *transcend from concrete (***C***) to pictorial (***P***) to visualization ***(V***) and then abstract ***(A***)*.

The formation and use of mathematics scripts for arriving at addition facts provides a student practice in oral temporal processing, therefore, their oral working memory is improved. On the other hand, the use of concrete materials such as: TenFrames, Visual Cluster cards, and Cuisenaire rods because of their patterns, color, shape, size, and organization help students to visualize the mathematical actions. This, in turn, improves visuo-spatial processing and working memory. These materials, therefore, not only improve students’ mathematics content mastery, but also working memory and other executive function components, and learning potential. When they feel successful and realize that they are able to find the answers without counting, they develop positive feelings toward mathematics and increased engagment.

With consistent practice, children master these facts (e.g., they have understanding of the concept/strategy, can derive fact accurately, can visualize the relationship/script, automatize the fact, and apply to another fact or problem solving).

All of the major strategies of deriving addition facts (finding the sums up to 20): are based on the above prerequisite skills:

(i) **M + N = N + M** (*commutative property of addition operation)*, (ii) ** 9 + N** (number), (ii)

Above strategies are aimed at developing the concept of addition, mastering addition facts, and at the same time, developing children’s learning ability to learn future concepts. In other words, the learning disability of a child is not a limiting condition forever. With proper methods, one can improve learning skills and manage the limitations, if they are not improved.

]]>When I reached middle school, the headmaster welcomed us and gave a little “talk” on what was expected of us in the middle school. He talked about forming the habit of reading everyday for pleasure and for school, importance of doing homework every night, selecting a sport that we could enjoy even after we left school, keeping a good notebook for classwork, writing everyday something of interest, become proud of our school, and then he said: “Those of you who have mastered your multiplication and division facts, you will be finishing eighth grade with a rigorous algebra course and then finish high school with a strong calculus course.” After laughter subsided, we realized the importance of the statement our elementary mathematics teacher–Sister Perpetua used to make as she was making sure that we had mastered our multiplication tables by the end of third grade and division facts by the end of fourth grade. We had heard about headmaster’s welcoming speech from her and the students who had gone before us. As headmaster and a demanding math teacher, he was very popular and respected by teachers, parents, and most students. He would repeat the ideas many times after that. It was more than sixty-five years ago, but his words are still fresh in my mind.

In my more than fifty-five years of teaching mathematics from number concept to Kindergarteners to pure and applied mathematics to graduate students (in mathematics, engineering, technology, and liberal arts), and preparing and training teachers for elementary grades to college/university, I am strongly convinced that no student should leave the fourth grade without mastering multiplicative reasoning—its language, conceptual schemas/ models, multiplication and division facts, and its procedures—including the standard algorithms.

* A. Concept, Role and Place of Multiplication in the Mathematics Curriculum:* 1. After

2. Whereas, in the context of addition and subtraction, we could express and understand numbers in terms of comparions of smaller, greater and equal, with multiplication and division, numbers can be expressed in terms of each other and we begin to see the underlying structures and patterns in the number system. Multiplicative reasoning provides the basis of measurment systems and their interrelationships (converting from larger unit to smaller unit (you multiply by the conversion factor and vice-versa. It is the foundation of understanding the concepts in number theory and representations and properties of numbers (even and odd numbers; prime and composite numbers, laws of exponents, etc.), proportional reasoning (fractions, decimals, percent, ratio, and proportion) and their applications.

3. * The move from additive to multiplicative thinking and reasoning* is not always smooth. Many children by sheer counting can achieve a great deal of accuracy and fluency in learning addition and subtraction facts, and at least for some multiplicaion facts. However, it is not possible to acquire full conceptual understanding (the models of multiplication and division), accuracy (how to derive them efficiently, effectively, and elgantly), fluency (answering correctly, contextually, in prescribed and acceptable time period), and mastery of multiplicative reasoning by just counting.

**B. Definition**: Qualitatively and cognitively, for children, * multiplicative reasoning is a key milestone* in their mathematical development. It is a higher order abstraction: addition and subtraction are abstractions of number concept and number concept is an abstraction of coutning. Addition and subtraction are one-dimensional cocnepts and are represented on a number line. Multiplication and division, as abstractions of addiotn and subtraction, start out as one-dimensional (as repeated addition and groups of), but they become two-dimensional concepts/ operations (i.e., as an array and area of a rectangle representations). Lack of complete understanding and mastery of multiplicative reasoning can be a real and persistent barrier to mathematical progress for students in the middle years of elementary school and later. Compared with the relatively short time needed to develop additive thinking (from Kindergarten through second grade), the introduction, exploration, and application of ideas involved in multiplication may take longer. Understanding of multiplicative reasoning (i.e, the four models–repeated addition, groups of, an array, and area of a rectangle) is truly a higher order thinking as the basis of higher mathematics.

1. * The main objective *of the

**2.** ** The reasons for difficuties in mastering multiplication and multiplication tables**: The first real hurdle many children encounter in their school experience is mastering multiplication tables with fluency. Even many adults will say: “I never was able to memorize my tables. I still have difficulty recalling my multiplication facts.” It is a worldwide phenomenon. Everyone agrees that chidren should master multiplication tables, but there is disagreement in opinions about

At the time of evaluation for a student’s learning difficulties/disabilities/ problems, when I ask him/her, ‘Which multiplication tables do you know well?’ Inevitably, the reply is ‘The 2’s, 5’s and 10’s.’ Some of them would add on the tables of 1’s, 0’s and 11’s to their repertoire. If I follow this up by ever so gently asking the answer for 6 × 2, then the response is: “I do not know the table of 6.” On further probing, I get the answer. Most frequently, the student finds the answer by counting on fingers 1-2, 3-4, 5-6, 7-8, 9-10, 11-12. 6 × 2 is 12. Some will say: 6 and then 7, 8, 9, 10, 11, and 12. 6 × 2 is 12. All along, the student has been keeping track of this counting on his/her fingers. Another way the answer is obtained by reciting the sequence: 2, 4, 6, 8, and 12. Here also the record of this counting is kept on his/her fingers. Both of these behaviors are indicative of lack of mastery of multiplication facts. They are also indications of the child having ** inefficient strategies **for arriving at multiplication facts. Skip counting forward on a number line or counting on fingers is not an efficient answer to masering multiplication facts.

Similarly, during my workshops for teachers, when I ask them to define “multiplication.” Most people will define multiplication as “repeated addition,” which is something that most of them know about multiplication from their school experience. Then I ask, according to your definition, what do you think the child would do to find 3 × 4? What does that mean to the child? The answer is almost immediate. “It means that child thinks 3 groups of four. He would count 4 three times.” As we can see, the person is mixing the two models of multiplication: “*repeated addition*” (3 repeated 4 times: 3 + 3 + 3 + 3 and “*groups of*” (4 + 4 + 4). Their definition and the action for getting the answer do not match. There is incongruence between their *conceptual* *schema* for the concept of multiplication and the *procedure for developing a fact*. Many children when deriving multiplication facts have the same confusion. To derive 6 × 8, A child would say (if he knows the table of 5–a very good sign): “I know 8 × 5 is 40 and then I add 6 so the answer for 6 × 8 = 46. The reason for wrong answer is this confusion in mixing the definitions. Children should understand* different definitions of multiplication. *The concept and problems resulting in multiplication emerge in several forms

On the other hand, repeated adddition and array model are limited to whole number multiplication. And, groups of model is helpful in conceptualizing the concept of multiplication of fractions and decimals. Children also acquire the misconception that “multiplication makes more” when they are exposed to only repeated addition and the array model. In such a situation, I say to them: “you are right. But what happens when you have to find the product of two fractions ½ × ⅓? What do you repeat how many times? The answer, invariably is: “You cannot. You multiply numerator times numerator divided by denominator times denominator.” Or, “what do I repeat when I want to find 1.2 × 1.3?” At this time, most teachers will give me the procedure of multiplying decimals. “Multiply 12 and 13 and then count the number of digits after the decimal.” If I pursue this further by asking: “How do we find the product (a + 3) (a + 2)?’ I begin to loose many in my audience. If, a person has complete understanding of the concept of multiplication, they can easily extend the concept of multiplication from whole numbers to fractions, decimals, and algebraic expressions. Only, the models “groups of” and the “area of a rectangle” models help us conceptualize the multiplication of fractions, decimals, integers, and algebraic expressions. And, only the area of a rectangle model helps us to derive the standard procedure for: multiplication of fractions/decimals, binomilas, distributive property of multiplication of arithmetic and algebraic expressions.

As one can see from this exchange, according to most teachers, the model or definition for conceptualizing multiplication changes from grade to grade from person to person. Rather than understanding the general principle/concept of multiplication, students try to solve problems by specific or ideosyncratic methods. Later, they find it difficult to conceptualize schemas/models/procedures for different examples of multiplication problems (with different types of numbers) and they give up. For example, they have difficulty reconciling the multiplication of fractions and decimals with their intial schema for multiplication (repeated addition or array andd even groups of, in some situations). We beleive, they should be exposed to and should be thoroughly familiar to the four models of multiplication before we introduce them to procedures. They should practice mastering multiplication tables when they have learned and applied these four models of multiplication. Then, they can accomodate different situations of multiplication into their schema of multiplication and create generalized schema for multiplication. The most generalized model for multiolicaiton is the area of rectangle.

Some of the difficulties children have in learning the concept of multiplication are the result of the lack of understnading of these different schemas and the emphasis on sequential counting in teaching multiplication in most classrooms. Students are not able to organize them in their heads, see the connections between them, and the importance of learning these models. They also think that different number types (whole numbers, fractions, decimals, integers, rational/irrational, algebraic expressions, etc.) have different definitions of multiplications. They do not see that the definitions and models should be generalizable.

3. * Another reason for the difficulty* is the teaching of multiplication: Children

4. ** Mastering a concept** means, the student has the

From the outset, we want to emphasize that it is important for children to learn (understand, have efficient strategies for arriving at the facts, accuracy, fluency, and then automatization) their multiplication tables. Eventually, by deriving the facts using efficient strategies and applying them to problems, they will be able to recall multiplication facts rapidly (*8 times 3*? *Twenty-four*!), and then use this knowledge to give answers to division questions (24 ÷ 3? Eight!); use these multiplication and division facts to do long multiplications and divisions; and use them appropriately in solving problems. When the concept of multiplication is understood, then one should introduce division concept and help them see that multiplication and division are inverse operations. Cyisenaire rods are the best material for making this relationship clear. (See *How to Teach Multiplication and Division*, Sharma 2018).

C. **Transition from Addition to Multiplication**: * Pre-requisite Skills for Multiplication and Multiplication Tables: * 1.

**2. Additive Reasoning pre-requisite Skills for learning and masrering Multiplication Tables: **(a) 45 sight facts of adddition, (b) Making ten, (c) Making Teens’ numbers, (d) What is the next tens, (e) Adding multiples of Tens to a two-digit number (e.g., 27 + 30 = ? 59 + 50 = ? 40 + 10 =?), (e) Commutative property of addition, (f) Counting forward and backward by 1, 5, 10, and 2 from any number.

3. **The Order of Teaching Multiplication**** Tables**: Derivation of multiplication facts/tables is easier when the four models:

(i) * Commutative property of multiplication*: This reduces the work of deriving 100 facts to only 55, an easier task.

(ii) * Table of 1 *(19 facts), (iii)

(vi) ** table of 9** [11 new facts] The table of 9 has several clear patterns hidden in it. Children need to see them. For example, (a) the sum of the digits in the table of 9, from the facts we already know (from tables of 1, 10, 5, and 2) is always 9: 9 × 1 = 9 = 09, 0 + 9 = 9; 9 × 2 = 18, 1 + 8 = 9; 9 × 5 = 45, 4 + 5 = 9; 9 × 10 = 90; 9 + 0 = 9; (b) the tens’ digit in the table of 9 is 1 less than the number being multiplied with 9, 9 × 1 = 9 = 09, 1 – 1= 0; 9 × 2 = 18, 2 – 1 = 1; 9 × 5 = 45, 5 – 1 = 4; 9 × 10 = 90, 10 – 1 = 9. Let us, therefore, apply these two patterns to derive 9 × 7 = ? We use the two patterns: here in the ten’s place will be 7 – 1 = 6, and, then to make the sum of the two digits as 9, we know that 6 + 3 = 9, thus, 9 × 7 = 63, and by commutative property of multiplication, we have 9 × 7 = 7 × 9 = 63. This process helps children to easily memorize the table of 9. We can also derive the fact 9 × 7 in several other ways: (a) by using the distributuve property of mulitplication over subtraction: we already know that , 10 × 7 = 70 ; 9 × 7 = (10 – 1) × 7 = 10 × 7 – 1 × 7 = 70 – 7 = 63; (b) using distributive property of multiplication over addition, 9 × 7 = 9 × (5 + 2) = 9 × 5 + 9 × 2 = 45 + 18 = 45 + 20 – 2 = 65 – 2 = 63, Or, 9 × 7 = 9 × 5 + 9 × 2 = 45 + 18 = 45 + 10 + 8 = 55 + 5 + 3 = 60 + 3 = 63.]

(vii) ** Table of 4 **(9 new facts). Since 4 is double of 2, the entries ib the table of 4 are double of table of the corresponding entries in the table of 2. For example, 4 × 7 = 2(2 × 7) = 2 × 14 = 2 × 10 + 2 × 4 = 20 + 8 = 28; Or, 4 × 7 = 4 (5 + 2) = 4 × 5 + 4 × 2 = 20 + 8 = 28. Or, 4 × 7 = (2 + 2)7 = 2 × 7 + 2 × 7 = 14 + 14 = 28.

(viii) * Remaining facts*: The total number of multiplication facts derived so far: 19 + 17 + 15 + 13 + 11 + 9 = 84. The remaining 16 facts are: 3 × 3; 3 × 6, 6 × 3; 3 × 7, 7 × 3; 3 × 8, 8 × 3; 6 × 6; 6 × 7, 7 × 6; 6 × 8, 8 × 6; 7 × 7; 7 × 8, 8 × 7; and 8 × 8. And, because of the commutative proeprty of multiplication, the number is reduced to 10. These 10 facts can be mastered by children in a week. These remaining facts should be derived by decompositon/ recomposition. For example, let us consider: 8 × 6 = ?.

5. * Improving Times Table Fluency*:

In the light of many similar studies, concept-based instruction involving efficient and effective methods that can be generalized and uses pattern-based continuos materials (Cuisenaire rods, Visual Cluster cards, etc.) that help in developing the script are better. Once children know the tables of 1, 2, 10, and 5 and can derive the other facts by using effective scripts, they should paractice the tables with games. We have found the following games using Visual Cluster cards to be very effective.

**Game Four: Mastering Multiplication Facts**

*Materials:** *A deck of Visual Cluster Cards (Playing cards without numbers) without face cards or with face cards. Each face card is, intially, given a fixed value (Jack = 2, Queen = 5, and King =10), later they are given values as: Jack = 11, Queen = 12, and King =15).

*How to Play**: *

- The whole deck is divided into two to four equal piles (depending the number of players).
- Each child gets a pile of cards. The cards are kept face down.
- Each person displays two cards face up. Each one finds the product of the numbers on the two cards. The bigger product wins. For example, one has the three of hearts and a king of hearts (value 10), the product is 30. The other has the seven of diamonds and the seven of hearts, the product is 49. The second player wins. The winner collects all cards.
- If both players have the same product, they declare war. Each one puts down three cards face down. Then each one turns two cards face up. The bigger product of the two displayed cards wins. The winner collects all cards.
- The first person with an empty hand loses.
- Initially, the teacher or the parent should be a player in these games. Their role is not only to observe the progress, mediate the disputes, keeping pace of the game and encouragement, but also to help them in deriving the fact when it is known to a child. For example, if the child gets the cards: 8 of diamond and 7 of spade. Teacher asks: What is the multiplication problem here? “8 × 7” The teacher asks: Do you know the answer? “No” Which is the bigger number? “8” Can you break the 7 into two numbers (point ot the clusters of 5 and 2 on the 7-card)? “5 and 2” If the 7-card was 5-card, then the problem would be 8 × 5. If the 7-card was 2-card, then the problem would be 8 × 2. Now, 7-card has 5 and 2, so the problem is: Is 8 × (5 + 2). Is 8 × 7 is same as 8 × (5 + 2). “Yes!” So, 8 × 7 = Is 8 × 7 = 8 × (5 + 2) and is made up of two problems: 8 × 5 and 8 × 2. What is 8 × 5? “40” What is 8 × 2? “16” Now, What is 8 × 5 and 8 × 2 together? “40 + 16” What is 40 + 16? “56” Good! What is, then, 8 × 7? “56.” All this should be done orally.

In one game, children will derive, use, and compare more than five hundred multiplication facts. Within a few weeks, they can master multiplication facts. Once a while, as a starting step, I may allow children to use the calculator to check their answers as long as they give the product before they find it by using the calculator.

*Game Five: Division War*

** Objective: **To master division facts

*Materials:** *Same as above

** How to Play: **Mostly, same as above.

- The whole deck is divided into two to 4 equal piles (depending on the players.
- Each child gets a pile of cards. The cards are kept face down.
- Each person displays two cards face up. Each one finds the quotient of the numbers on the two cards. The bigger quotient wins. For example, one has the three of hearts and a king of hearts (value 10). When 10 is divided by 3, the quotient then is 3 and 1/3. The other has the seven of diamonds and the seven of hearts, the quotient is 1. The first player wins. The winner collects all cards.
- If both players have the same quotient, they declare war. Each one puts down three cards face down. Then each one turns two cards face up. The bigger quotient on the two displayed cards wins. The winner collects all cards.
- The first person with an empty hand loses.

In one game, children will use more than five hundred division facts. Within a few weeks, they can master simple division facts. I allow children to use the calculator to check their answers as long as they give the quotient before they find it by using the calculator.

*Game Six: Multiplication/Division War*

** Objectives: **To master multiplication and division facts

*Materials:** *Same as above

** How to Play: **Almost same as the other games

- The whole deck is divided into two to four equal piles (depending on the number of players.
- Each child gets a pile of cards. The cards are kept face down.
- Each person displays three cards face up. Each one selects two cards from the three, multiplies them, and divides the product by the third number (finds the quotient of the numbers). The bigger quotient wins. For example, one has the three of hearts, the seven of diamonds, and a king of hearts (value 10). To make the quotient a big number, the player multiplies 10 and 7, gets 70, and divides 70 by 3. The quotient is 23 1/3. The other player has the seven of diamonds, the seven of hearts, and the five of diamonds. He/she decides to multiply 7 and 7, gets 49, divides 49 by 5, and gets a quotient of 9 4/5. The first player wins. The winner collects all cards.
- If both players have the same quotient, they declare war. Each one puts down three cards face down. Then each one turns three cards face up. The bigger quotient on the three displayed cards wins. The winner collects all cards.
- The first person with an empty hand loses.

In one game, children will use more than five hundred multiplication and division facts. They also try several choices in each display as they want to maximize the outcome. This teaches them problem solving and flexibility of thought. Within a few weeks, they can master simple division facts. I allow children to use the calculator to check their answers as long as they give the quotient before they find it by using the calculator.

^{[1]}See previous posts on *Numbersense*; *Sight Facts and Sight Words*; *What does it Mean to Master Arithmetic Facts?, etc. *

^{[2]}See previous posts on *Non-Negotiable Skills at the Elementary Level*. For a fuller treatment on the topic see: ** How to Teach Multiplicative Reasoning** by Sharma (2019).

To achieve this goal of quantitative domain, * at the end of Kindergarten*, a child should have mastered: (a)

Mastery of ** number concept** is the foundation of arithmetic. The ten numbers/digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are the

Similarly, ** in the quantitative domain, at the end of second grade,** a child should have mastered: (a)

* In the quantitative domain, the focus of three-years of mathematics instruction*, from Kindergarten to second grade, is that by the end of second grade, children have mastered the concept of Additive Reasoning. Aquiring

The concept of * mastery *of mathematics concept/skill/procedure means: (a) the child possesses the appropriate

A ** strategy** is appropriate if it

Learning and mastering arithmetic facts is dependent on three kinds of pre-requisite skills: (a) * Mathematical*:

Before and during the game, the emphasis should be on the use of sight facts, decompositon/recomposition, making ten, and making ten. During first few games, a teacher/parent should participate in the game with the children. During the game she should develop script fot finding the facts. For example, if during the * Addition War Game *game a child gets two cards: 6 of diamonds and 8 of spades, he needs to find the sum 6 + 8. If the child does not know the answer readily, then the teacher/parent should practice fidning the sum of 8 + 6 by asking questions that help the child practice the necessary pre-requisite skills and develop a script that will help the child to find the answers independently. For example, (a) Look at your cards: (points the cards: 8 and 6).

All of these questions, with the help of visual cluster cards (Cuisenaire and Empty Number Line), should be answered and practiced orally. This process develops many of these pre-requisite skills individually and then helps integrate them. For example, working with the patterns on the Visual Cluster cards and then visualization of the cards aids in the development of the working memory. The organized sequential script helps them focus, organize and develop deductive reasoning. The reorganizing the pattern on the first card into sub-patterns and then integrating them with the patterns of the second card helps with the acqusition of decomposition and recomposition skills. The game setting: playing the game involves practicing these skills again and again and soldiifes these skills. For example, in playing the Number Addition War involves making, hearning, and practiicing more than 500 addition facts. Neurologically, questions instigate neural firing and making connections, that in turn invites oligodendrocytes–(oligo) to instigate the production of myelin–creating covering around the nerve fibers, that in turn controls and improves the impulse, and the impulse speed is skill. Each time a child practicies the script, the nerve fibres get stronger and wrapping wider and wider making the pathway of the nerve impulses into a major “highway.” The integration of (a) practicing the script, (b) visualizing the action guided by the script, (c) acceleration of the neural firing (better myelination), (d) and reducing the refractory time (the wait required between one signal and the other) makes learning optimal. The increased speed abd decreased refractory time No child will practices the number examples in a formal setting as he practices in one game. With the Number Addition War Game, children master their addition facts in a very short time. And that too with great deal of pleasure.

To make the learning * robust* and making children

Display two Visual Cluster cards: 8 of dimonds and 6 of clubs. Do you know what addition problem can you make form these numbers? “8 + 6 or 6 + 8.” Good! What is 8 + 6? “14.” How did you find the answer? “8 + 2 is 10 and then 4 more is 14. So, 8 + 6 is 14.” What about 6 + 8? “14.” How did you know that quickly? “Because 8 + 6 = 6 + 8.” What property is that? “Turn-around-fact.” What is another name for that property? “Commutative Property of Addition.” Is there any way you can find 6 + 8? “I do not know.”

Display two Visual Cluster cards: 6 of dimonds and 6 of clubs. Do you know what addition problem can you make form these numbers? “6 + 6.” Good! What is 6 + 6? “16.” How did you find the answer? “6 + 4 is 10 and then 2 more is 12. So, 6 + 6 is 12.” What about 6 + 8? “14.” How did you know that quickly? What peoperty is that? Doubles property.” Good!

Display two Visual Cluster cards: 8 of dimonds and 8 of clubs. Do you know what addition problem can you make form these numbers? “8 + 8.” Good! What is 8 + 8? “16.” Can you find 8 + 6 using the fact that 8 + 8 = 16? “I do not know.” Is 8 + 6 is less than 8 + 8 or more than 8 + 8? “It is less.” If, the child begins to count. The teacher/parent should intervene. Look at the second 8-card. If you cover the 2 from the card, what do you see on the card. “a 6.” What addition problem do you have now? “8 + 6.” Can you figure out the answer for 8 + 6? “Yes, it is 14.” How do you know? “I know 8 + 6 = 14.” So, 8 + 6 is how much les than 8 + 8? “2 and 8 + 6 = 8 + 8 – 2.” Good!

Display two Visual Cluster cards: 8 of dimonds and 6 of clubs. Do you know what addition problem can you make form these numbers? “8 + 6 or 6 + 8.” Good! What is 8 + 6? “14.” How did you find the answer? “8 + 2 is 10 and then 4 more is 14. So, 8 + 6 is 14. Or, 6 + 6 + 2 = 14. Or, 8 + 8 – 2 = 14.” Do you know any other way? “I do not think so!” What if you took the one pip from the 8-card an put it on the 6-card, what problem would you have? “7 + 7.” What is 7 + 7? “14.” How do you know? Doubles property. Great! Can you apply making 10 strategy to this problem? “Yes! 7 + 3 is 10 and 10 + 4 = 14.” Great! Now, you know several ways of finding 8 + 6 or 6 + 8. How far apart are 8 and 6? “2 apart.” What number is between 6 and 8? “7.” So, 6 + 8 is same as 7 + 7. **When two numbers are 2 apart, their sum is double of the middle property. **

Practicing multiple strategies for finding the answer improves a child’s cognitive potential. They begin to see more realtionships, patterns, and concepts. They do not get helpless when they do not have the answer. They take action. This is an anti-dote to math anxiety.

*Game Three: Number Addition War*

** Objective:** To master addition facts

*Materials:** *Same as above

*How to Play: *

- The whole deck is divided into two equal piles of cards.
- Each child gets a pile of cards. The cards are kept face down.
- Each person displays two cards face up. Each one finds the sum of the number represented by these cards. The bigger sum wins. For example, one has the three of hearts (value 3) and a 10 or a king of hearts (value 10). The sum of the numbers is 13. The other person has the seven of diamonds (7) and the seven of hearts (7). The sum is 14. The person with the sum of fourteen wins. The winner collects all the displayed cards and puts them underneath his/her pile.
- The face card and the wild card can be assigned any number value up to ten.
- If both players have the same sum, there is war. For example, one has the five of hearts (value 5) and the seven of clubs (value 7), and then the sum is 12. The other person has the six of spades (value 6) and the six of clubs (value 6). They declare war.
- Each one puts three cards face down. Then each one displays another two cards face up. The bigger sum of the last two cards wins.
- The winner collects all the cards and places them underneath his/her pile.
- The first person with an empty hand loses.

This game is appropriate for children who have not mastered/automatized addition facts.

Initially, children can count the objects on the cards. However, fairly soon they begin to rely on visual clusters on the cards to recognize and find the sums. In one game, children will encounter more than five hundred sums. Within a few weeks, they can master all the addition facts. Initially, if the child does not know his sight facts, the game can be played with dominos or with a deck of Visual Cluster cards with numbers only up to five. Then, include other cards.

I sometimes allow children to use the calculator to check their sums. The only condition I place on calculator use is that they have to give the sum before they find it using the calculator. Over a period of time, calculator use declines and after a short while, they are able to automatize the arithmetic facts. After they have learned the 10 ×10 arithmetic facts (sums up to 20), you can assign values to the face cards: Jack = 11, Queen = 12, and King = 13. The joker has a value assigned by the player. Its value can be changed from hand to hand. The joker is introduced with a variable value so that children can get the concept of variable very early on.

** Variation 1: **After a while, you might make a change in the rules of the game.

Each child displays three cards, discards a card of choice, and finds the sum of the remaining two cards.

** Variation 2: **Each child displays three or four cards, finds the sum of the three or four cards, and the bigger sum wins.

*Game Three: Subtraction War*

** Objective: **To master subtraction facts

*Materials:** *Same as above

*How to Play: *

- The whole deck is divided into two equal piles of cards.
- Each child gets a pile of cards. The cards are kept face down.
- Each person displays two cards face up. Each one finds the difference of the two cards. The bigger difference wins. For example, one has the three of hearts and a king of hearts (value 10), and then the difference is 7. The other has the seven of diamonds and the seven of hearts, and then the difference is 0. The first player wins. The winner collects all cards.
- If both players have the same difference, they declare war. Each one puts down three cards face down. Then each one turns two cards face up. The bigger difference of the two displayed cards wins. The winner collects all cards.
- The first person with an empty hand loses.

As in addition, children can initially count the objects on the cards. However, fairly soon they begin to rely on visual clusters to recognize and find the difference. In one game, children will use more than five hundred subtraction facts. Within a few weeks, they can master subtraction facts. Initially, the game can be played with dominos.

I allow children to use the calculator to check their answers. As mentioned above, the only condition I place on calculator use is to give the difference before they find it using the calculator. Over a period of time, calculator use declines and after a short while, they are able to automatize the arithmetic facts. This game is appropriate for children of all ages to reinforce subtraction facts.

** Variation 1: **After a while, you might make a change in the rules of the game. Each child displays three cards, discards a card of choice, and finds the difference using the remaining two cards.

** Variation 2: **Each child displays three cards, finds the sum of any two cards, and subtracts the value of the third card. The bigger outcome of addition and difference wins.

**Variation 3** :Each child displays three or four cards, an objective number is decided and finds the result by adding or subtracting of any combination of cards gets the declared number as the result. The bigger outcome of addition and difference wins. No number can be used more than once.

Children, all over the world, love to play games. I have successfully used games for initial teaching and remedial mathematics instruction, particularly, for learning arithmetic facts (addition, subtraction, multiplication, and division), comparison of fractions, and comparing and combining integers. An ordinary deck of playing cards, a pair of Dice and Dominos are good tools for teaching arithmetic, particularly, number conceptualization and simple arithmetic facts. However, using cards from an ordinary deck assume number concept, in thier use and Dominos and Dice only teach * subitizing*. Whereas, A set of Visual Cluster Cards helps children to learn all the components of Nubmer Concept: learning

Playing cards are used for playing games all over the world. Every culture has developed playing cards and games related to them. The games and their complexity vary from simple to complex and from simple comparison to strategies. The number and type of games played using an ordinary deck of playing cards abound. Games, using playing cards, are enjoyed by all—from children to adults.

One of the most popular games children play is called the ** Game of War**. This is a family of card games. These games, under various names, are played by children and adults all over the world. I have adopted many of these games for teaching mathematics concepts and reinforcing them. I call these games: ‘

I have designed a special set of playing cards called: *Visual Cluster Cards*^{TM} for playing these games. Visual Cluster Cards are without numbers on them. When children use these cards, within few days, they learn the most important component of number: decomposition/ recomposition. Through decomposition/ recomposition, they acquire the 45 ** sight-facts** (addition and then subtraction facts of numbers up to 10). Visual Cluster Cards are better suited for these games

Visual Cluster Cards are modified ordinary deck of cards, in their design and in number. They are of two types: With face cards and without face cards. Both types of Visual Cluster cards have several arrangements of clusters for numbers such as: 0 (one blank card), 3 (two clusters), 8 (two), 9 (four), wild card (two), and 10 (two). All other numerals (1, 2, 4, 5, 6, and 7 have one card in each suit (spade, club, diamond, and heart). There are 60 cards (the deck without face cards) in this deck. The other deck of Visual Cluster cards includes face cards, in addition to all the other cards. The blank card represents zero and the wild card as a variable–assuming the value the context and the player assigns. In this deck, all face cards represent 10 (a good option when working on numbers up to 10) or the jack represents 11, queen represents 12, and king represents 13.

In both decks, the cluster of objects (pips, icons) represents the ** numeral **and the color (black = positive, red = negative) of the pips represent the

Children learn the quantity (numeral), number (positive and negative numerals) represented by the cards by observation (by sight), ultimately without counting. Since, children derive and learn the relationship between numbers up to ten by sight, these facts are called ** sight facts**.

The ordinary Game of War is played by children all over the world. My game begins in the same way as the Game of War. It is played essentially the same way and is easy to learn. Before, they play the game, however, it is important that children become familiar with the deck of Visual Cluster cards, particularly, the patterns of visual clusters on each card.

Visual Cluster cards have clusters of objects displayed on the card. For example, there are five diamonds displayed in the middle in a particular pattern–a pattern that encourages decomposition/recomposition (see below).

An arrangement of this type is called a *visual cluster*^{[4]}. The particular arrangement above is the visual cluster for five. It will be called the numeral 5 up to fifth grade. Later, it will be called numeral 5 and number ^{+}5.

Because of the patterns of pips, on individual visual cluster cards, they can be recognized, without counting, visualized, and then committed to memory with ease. The special nature of the visual pattern of a cluster of pips, representing the quantity, on a card helps a child to form a vivid image of that quantity, therefore, the numeral/quantity represented by the card, in their minds. Each Visual Cluster card is organized according to a particular cluster. This helps players to recognize the size of collections (up to 10) without counting. This also helps children to integrate: (a) ** orthographic image** (5) of the numeral (when it is formalized in writing), (b) the

Children who are not able to form and hold these clusters in their minds and are, therefore, unable to recognize the size of a collection of objects by observation, have not conceptualized number, yet. This lack of integration of these three elements is a symptom and the manifestation of dyscalculia. Research supports this observation and shows that, in such a case, children have difficulty in learning number concept, number relationships, particularly addition and subtraction facts and other higher concepts, and later operations on numerals and numbers (i.e., integers, etc.). These children keep on counting on fingers or on number line to find the sums and differences of even two small numbers. They also have great difficulty in automatizing arithmetic facts.

The following games not only help children to conceptualize number but also help them to master arithmetic facts. These games are highly motivating to children.

There are several games, in this series, that are variations of other popular card games, such as ** “Go Fish.”** If you use or are aware of any card games that relate to number and number relationships, I would love to hear about them (maheshsharma@me.com).

**Game One: Visual Clustering and Comparison of Numbers**

*(For children age 3 to those who are having difficulty mastering arithmetic facts)*

** Objective:** To teach number concept—numberness, decomposition/ recomposition, and sight facts.

The game can be played between two or three players. However, it is most effective between two players.

** Materials:** Take a deck of Visual Cluster cards including jokers (joker can assume any number value, according to context). In the case of Visual Cluster Cards with face cards, each card’s value is the number of objects displayed by the visual cluster on the card (e.g., Ace = 1 and the blank card = 0). For example, the four of diamonds, clubs, spades or hearts will be known as number/numeral four.

Each face card, jack, queen, and king is initially given the value of ten. The ace represents number one. The joker can assume any value and can be different each time it is used. When children know the teen’s numbers, then you can introduce: jack = 11, queen = 12, King = 13.

*How to Play: *

- The whole deck is divided into two equal piles of cards (if two players).
- Each child gets one pile of cards. One can also distribute the cards equally by counting out loud (This teaches children sequence of numbers and their location on the sequence of numbers. This increases number vocabulary–lexical enries for number) . Each person keeps the cards face down.
- When the game begins, each person turns a card face up. The bigger value card wins. For example, one has the three of hearts (value 3), and the other person has the seven of diamonds (value 7). The seven of diamonds wins. The winner collects all the displayed cards and puts them underneath his/her pile. (When playing this game with integers, three of hearts represents -3 and 7 of diamonds will represent -7).
- If both players have the same value cards (for example, one has the five of hearts, and the other has the five of spades), they declare war: “I declare war.”
- Each player puts three cards face down on each sound of the word, in succession, saying I (for the first card) declare (for the second card), and war (for the third card). Then each player displays a fourth card face up. The bigger valued fourth card wins. If they match again, the same process is repeated.
- The winner collects all cards and places them underneath his/her pile.
- The first person with an empty hand loses.

This game is appropriate for pre-K, Kindergarteners, and other children who have not mastered number concept. Number conceptualization is dependent on five interconnected skills: (i) Having a *large number vocabulary*, (ii) *one-to-one correspondence* with sequence, (iii) *visual clustering* (extension of *subitizing*)—recognizing a cluster of objects up to five by observation (without counting) is called *subitizing* and recognizing up to 10 objects is called *visual clustering*, (iv) *decomposition/recomposition*, and (v) *ordering*. This game develops all of these prerequisite skills and many more. Children with a lack of number concept have great difficulty in learning arithmetic facts and can derive them only by sequential counting. Which is a very inefficient strategy. Initially, for a short while, children can count the objects on the cards. However, fairly soon they begin to rely on visual clusters to recognize the value of cards. In a game, children have the opportunity of comparing almost five hundred pairs of numbers.

*Game Two: What Makes This Number*

*(For children age 3 to those who are having difficulty mastering arithmetic facts)*

** Objective:** To master addition sight facts

*Materials:** *Same as above

*How to Play: *

- The whole deck is divided into two equal piles of cards.
- Each child gets a pile of cards. The cards are kept face down.
- Each person displays one card face up. Each one finds two numbers whose sum is their card. For example, one has the three of hearts (value 3) and, therefore, gives two sight facts: 1 + 2 = 3, 2 + 1 = 3. The other has the seven of diamonds, the sight facts are: 1+ 6, 2+ 5, 3 +4, 4+3, 5+2, 6+1. The one with more sight facts wins. If the child, with the bigger number, cannot produce all the sight facts, the other player gets a chance and if he/she can give all the sight facts, he/she wins. In general, the person who is able to produce all the sight facts correctly and has the bigger number wins. The winner collects all the displayed cards and puts them underneath his/her pile.
- If both players have the same number of sight facts, there is war. For example, one has the five of hearts (value 5) and gives all the sight facts and the other has five of clubs (value 5) and gives all the sight facts. Or, one has five of diamonds and gives three sight facts only, and the other has nine of clubs (value 9) and gives three sight facts only, they declare war.
- Each one puts three cards face down. Then each one displays another card face up. The bigger number of sight facts wins.
- The winner collects all the cards and places them underneath his/her pile.
- The first person with an empty hand loses.

This game is appropriate for children who have not been introduced to sight facts or have not mastered/automatized simple addition facts.

Initially, children will count the objects on the cards. However, fairly soon they begin to rely on visual clusters to recognize and find the sums. Within a few weeks, they can master all the 45 sight facts^{[5]}. Initially, the game can be played with dominos or with a deck of cards of numbers up to five.

This series of posts will continue. In future editions, number games relating to other operations (inlcusding algebraic operations) will be included. Next few games will be on arithmetic operations.

^{[1]}Copyright 2008 with Mahesh Sharma.

^{[2]}Visual Cluster Cards are available from Center for Teaching/Learning of Mathematics ($15 per deck plus $4.00 for shipping and handling).

^{[3]}Number Conceptualization by Sharma (2008).

^{[4]}Same as above.

^{[5]}The list of sight facts and how to teach them is included in How to Teach Number Concept Using Visual Cluster cards (Sharma, 2017). Also see the post on ** Sight Words and Sight Facts** on this Blog.