Mathematics Education Workshop Series with Professor Mahesh Sharma Academic Year 2019-2020

There are still seats available for the upcoming workshop Proportional Reasoning: How to Teach Fractions Effectively (Part II): Concept and Addition and Subtraction (for grade 3 through grade 9 teachers) on Friday, February 28th! Scroll down to learn more! According to Common core State Standards in mathematics (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. 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 $59.00 per workshopand 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. REGISTER!!!

Dysgraphia, Agraphia, and Related Math Difficulties

Clear, succinct, and accurate writing achieves two objectives:

to communicate and to be understood clearly and accurately.

Mahesh C. Sharma

The ability to write is a fundamental component of literacy and numeracy, and is crucial for success not only in school but also in most workplace environments. Writing serves two purposes in learning; receiving information efficiently and communicating accurately, effectively, and comprehensively. For both, the writer and the reader, it should bring understanding. To be understood is fundamental to human communication. Writing is an important and complex task that typically develops in early childhood. There is strong evidence that the presence of a word’s written form leads to improved learning of its spelling and its spoken form. There is also evidence that writing a word may lead to better learning of its meaning. A small number of studies have also shown that the presence of a word’s written form benefits vocabulary learning in children with developmental language disorder, autism, Down syndrome, and reading difficulties. Unfortunately, today, a significant proportion of our students have poor handwriting and writing skills and some of them suffer from developmental dysgraphia. However, some of this situation exists because we do not help children to write enough, often, and properly.

Dysgraphia is a learning disorder in which the individual’s writing skills are below the level expected for his or her age and cognitive level. There are a variety of mechanisms by which dysgraphia may occur. It can present in isolation or with other learning, neurological, or psychiatric disorders, and can often go undiagnosed. Generally, the diagnosis and management of dysgraphia usually occurs in an educational setting.

Dysgraphia, from the Greek “dys” meaning “impaired” and “graphia” meaning “making letter forms by hand,” is a disorder of writing ability. Dysgraphia, thus, is an impairment in or lack of development of the process of writing and acquisition of effective writing skills. In some children, it is comorbid with language, reading, and numeracy disorders. And, that, in turn affect a person’s ability to write. Agraphia, on the other hand, is a loss of ability to write, usually because of insult or injury to the specific areas of the brain. In terms of outcome behaviors, however, the terms agraphia and dysgraphia are synonyms.

At its broadest definition, dysgraphia can manifest as difficulty writing at any level, including letter illegibility, slow rate of writing, difficulty spelling, and problems of syntax and composition. The term dysgraphia also refers to impaired spelling, whereas many researchers apply the label only to deficits affecting the motor planning or production processes required for handwriting. We include both perspectives within the scope of our discussion.

Although the writing difficulties begin during school-age years, they may not be manifested or recognized until later when the complexity of writing tasks increase. When considering this specific learning disorder, underlying other conditions that might be associated with learning difficulties should be carefully ruled out or their relationship understood. These may include intellectual disability, uncorrected visual or auditory acuity, other mental or neurological disorders, psychosocial adversity, lack of proficiency in the language of academic instruction, or inadequate educational instruction.

Dysgraphia is not identified as a disability or disorder in the Diagnostic and Statistical Manual of Mental Disorders, but it falls under the manual’s specific learning disorder category as an ‘impairment in written expression in different forms.’ The specifier of ‘with impairment in written expression’ includes deficits in spelling accuracy, grammar and punctuation accuracy, and clarity or organization of written expression.

Dysgraphia is a language-based learning difference that affects a student’s ability to produce written language—symbols, numbers, letters, and other representations, in different settings. In other words, it appears wherever alpha-numeric language or symbolic representations are used to communicate ideas in written form. In the early grades, students with dysgraphia may have difficulty with consistent letter and symbol (numbers and other mathematical/procedural symbols) formation, word and number spacing, aligning of digits—numbers and numerals, punctuation, and capitalization. In later grades, they may have difficulty with writing fluency, floating margins, writing mathematics (equations, fractions, exponents, etc.), and legible writing.

In a typical classroom, students with dysgraphia are often labeled ‘sloppy,’ ‘lazy,’ or ‘not detail-oriented’ by their teachers and parents instead of being correctly diagnosed with a learning disorder. Many students with dysgraphia, on the other hand, are often trying very hard, if not harder than others, just to keep up. Dysgraphia is an invisible disability that often goes hand in hand with dyslexia and dyscalculia. Like students with dyslexia, students with dysgraphia are often acutely aware of what they’re not capable of relative to their peers. Most people with dysgraphia make it as part of their identification: “I have poor handwriting, That is the way I am” that does not help either.

A. Dyslexia, Dyscalculia, and Dysgraphia

The clinical and scientific knowledge about developmental dyslexia has grown in the last several years. Whereas developmental dyslexia has moved into the focus of serious academic research, the investigation of developmental dysgraphia has garnered little attention from researchers and teacher trainers. And, the research and the impact of the interaction of dyslexia, dyscalculia, and dysgraphia is discussed even less. This is in spite of the incidences of comorbidity of these disorders, in significant numbers of children. For these disorders, there are different classifications and definitions in the literature, making it difficult for an average teacher to gain insight into the actual characteristics and causes of these disorders and their relationships and the impact of their interactions. And, more importantly, how to help these students.

In the diagnostic literature, the three disorders are described as specific learning disabilities (SLD) and further distinguished as “SLD with impairment in reading,” “SLD with impairment in written expression,” and “SLD with impairment in mathematics.” The three subgroups are categorized further as:

Specific learning disorder with impairment in reading can vary between problems in word reading accuracy, reading rate, or fluency and reading comprehension.

Specific learning disorder with impairment in written expression is divided into problems with either spelling accuracy, grammar and punctuation accuracy, and clarity or organization of written expression.

Specific learning disorders with impairment in mathematics is a conglomeration from numeracy (number concept, numbersense, and arithmetic operations), to algebraic and geometric concepts and operations to problems solving.

The difficulties and problems in the acquisition of writing skills and writing performance due to dysgraphia interfere with academic work and/or everyday activities. In this sense it is closely related to developmental dyslexia. Impact of dysgraphia doesn’t limit to words and writing—it also affects a students’ ability to learn, apply, and communicate mathematics skills. For instance, students with dysgraphia may have difficulty in learning place value, fractions, aligning numbers, organizing complex mathematics expressions and equations. Thus, because of this learning disorder the student has trouble learning in important school subjects: reading, math, or writing. It affects on performance in reading and mathematics, sometimes, exist despite adequate schooling, efforts, and other cognitive abilities.

Problems with handwriting can affect self-esteem, perception of ability, and relationships with peers and teachers. The prevalence of difficulties with writing depends on the definitions and parameters, however, researchers estimate the prevalence of developmental writing disorders to be about 7–15% among school-aged children, with boys being more affected than girls; 2–3 times. This percentage is similar to the incidence of developmental dyslexia, which is estimated to be about 6 to 17%. The incidence of dyscalculia in the school age population is about 6 to 8 percent. Problems with handwriting are a common reason for referral to occupational or physical therapy services.

However, the incidence of mathematics anxiety,acquired dyscalculia, and difficulty in learning of mathematics due to environmental factors (poor attendance, poor teaching, lackof practice, poor curriculum, poor standards of mastery, etc.), is much higher. It is a common refrain from, otherwise successful, individuals in academics and life that mathematics is difficult and they did poorly. At the same time, almost 40% of children with dyslexia also exhibit symptoms of dyscalculia. There are no figures for the comorbidity of dyslexia, dyscalculia and dysgraphia. An important diagnostic tool, estimates the prevalence of all learning disorders (including impairment in writing as well as in reading and/or mathematics) to be about 5–15% worldwide.

Some practitioners and researchers also relate dyslexia, agraphia and dysgraphia to lack of some specific perceptual and visual perceptual integration. It is believed, in some circles, that it may be connected with some vision issues also. For example, children with scotopic sensitivity syndrome is a genuine reason why some children do not enjoy reading or writing and present themselves as a case study for poor reading and poor handwriting, but the condition is not a cause of neither dyslexia nor for dysgraphia.

Many other children struggle with writing issues, but not all of them have dysgraphia. Some of them may have what I call as acquired dysgraphia (not to be confused with agraphia). Acquired dysgraphia —the poor handwriting may result because of lack of structure, organization, and poor academic work habits:

Poor equipment—improper writing paper (best student mathematics writing is done on graph paper), unsharpened, poor sized pencil, or doing mathematics with pen,

Poor motor and physical actions—poor pencil grip, lack of proper directions in the writing process (e.g., a right-handed person can draw a straight-line better if s/he draws it from left to right and top to bottom direction, otherwise the straight line will be difficult to achieve),

Poor organization and observation skills,

Lack of structure in problem solving (where to start, what to do first, what is unknown, what is known, how to make tables, charts, see patterns, draw line, recognize form, shape, and structures, etc.),

Lack of exposure to quality of instruction for handwriting or poor or lack of feedback, etc. For example, when during poor writing instruction, a child acquires poor habits of pencil grip, that child develops poor handwriting, lack of interest in writing, and the writing hand gets tired too easily during writing. These students produce the least amount of writing in the classroom.

Lack of practice in writing is another major reason for many children showing symptoms and behaviors that look like symptoms of dysgraphia. Many teachers never ask children to write much. They just give children sheets after sheets where they just fill numbers, symbols, or simple words. As a result, students do not have any practice in writing.

Trouble expressing oneself in writing or lack of ideas for writing are not part of the symptoms of dysgraphia. That may be due to many other factors. However, poor handwriting sometimes does inhibit children from writing. When a student has to focus and put so much effort on transcription, it can get in the way of thinking about ideas and how to convey them.

B. Dysgraphia and Mathematics

In the simple view of writing model, high-quality writing depends on appropriate fine motor skills (for transcription skills) and executive function: working memory (holding information from board or book or from thought), inhibition control (for focusing on text and place), organization (paying attention to the details of letter, number and symbol formation), and flexibility of thought (differentiating in size, placement, location, relative position and sizes of letters, numbers and symbols) —all of which can be difficult for children with dyslexia/dyscalculia/dysgraphia and, therefore, result in poor spelling and low overall writing quality.

This creates special challenges for children with dyslexia/dyscalculia. However, the training in developing efficient writing (and even reading) skills are good activities for improving the deficits in executive functions. This is particularly so when efficient and consistent instructional strategies integrating teaching mathematics content and effective writing skills are practiced. In mathematics, the writing plays more crucial role as the meaning of a mathematics expression may change by the size, location, and placement of a number and symbol.

Acalculia/dyscalculia, difficulty or inability to calculate, is a developmental disorder and may appear in conjunction with finger agnosia, agraphia/dysgraphia, and right-left discrimination impairment, which is known collectively as Gerstmann’s syndrome. This condition is generally caused by neurological or neurophysiological deficit. The Gerstmann syndrome or dyscalculia, dysgraphia, left-right confusion, and finger agnosia is generally attributed to lesions in the same focal area of the brain showing co-morbidity of the syndromes.

The agraphia/dysgraphia in Gerstmann syndrome can take the form of aphasic agraphia with errors in content of writing, apraxic agraphia manifested as a scrawl, or spatial agraphia seen as errors in management of positioning of letters on paper. A person may have aspects of apraxia (poorly formed letters) as well as aphasia (letter substitutions) in writing difficulty. The student may have difficulty organizing thought, work, and writing.

By developmental dysgraphia, in this article, we will mean difficulty in acquisition of writing (spelling, handwriting, or both), despite adequate opportunity to learn, and absence of obvious neuropathology or gross sensory–motor dysfunction.

Agraphia, on the other hand, is writing deficits resulting from brain damage, generally, in adulthood. Many experts view dysgraphia or agraphia as an issue with a set of skills known as transcription. These skills include handwriting, typing, and spelling. Since, writing process is complex and involves a diverse set of skills related to brain functions, and motor actions, there are several aspects of agraphia/dysgraphia.

Aphasic agraphia: Leaving out numbers, letters, and words in writing and components in compound figures or poorly made incomplete symbols, signs, numerals in copying mathematics problems from the board or book to their notebook or paper. It also means a child may have illegible handwriting: inconsistent sizes of letters, numbers, and mathematics symbols and inconsistent spacing between numbers and symbols.

Spatial agraphia: Errors in understanding spatial orientation/space organization, relative positions of objects (numbers, shapes, symbols, operation symbols) in written expressions, copying, writing, and executing (e.g., aligning multi-digit numbers, writing mixed fractions, algebraic operations, cannot align in addition and subtraction and place numbers in proper place in multi-digit operations (multiplication, and division, particularly in long-division). Distorted geometrical figures, relative sizes, and poor integration of figures, dimensions, and representations.

Apraxic agraphia: Poor handwriting might be caused by condition referred to as dyspraxia. Dyspraxia accompanies developmental coordination disorder (DCD). Developmental coordination disorder (DCD) makes it hard to learn motor skills and coordination. It’s not a learning disorder, but it can impact learning. Children with DCD struggle with physical tasks and activities they need to do both in and out of school. Therefore, apraxic agraphia affects writing symbols, signs, numerals. They end up writing as scrawls when copying problems and performing calculations. This results in confusion and errors in calculations, computations, and procedures (mixed fractions, exponents, and algebraic operations, etc.). They have difficulty copying numbers, shapes, figures, diagrams, and equations from the board.

Aphasia: Omits and substitutes letters, symbols, numbers, operational symbols in writing, copying and communicating mathematics information.

Acquired dysgraphia: Some children, as a result of poor handwriting instruction, lack of feedback on poor handwriting, poor standards handwriting, lack of practice in writing, and lack of discipline develop poor handwriting skills and develop distaste and helplessness in writing, therefore, they avoid tasks involving drawing or writing.

All of these dysfunctions manifest in difficulties, in form or other, with writing in mathematics, such as:

Forming letters, numbers, and symbols (i.e., in algebraic expressions and equations, they cannot differentiate between + and their letter t, or between ×, x, X, and x; for them they are the same);

Spacing letters, numbers, words, sentences, expressions, and calculations correctly and in an organized order and form on the page (i.e., one cannot trace the order of activity on the page);

Writing in a straight line (i.e, even when they are writing on a lined paper, dysgraphic students cannot write on the straight line, because they have never been instructed to respect a graph paper);

Making letters and numbers (fractions and exponents are written smaller than the whole numbers) according to the correct size (i.e., in algebra, they do not know the difference between capital letters and lower case letters);

Holding paper with one hand while writing with the other;

Holding and controlling a pencil or other writing tool;

Putting the right amount of pressure on the paper with a writing tool and when erasing; and,

Maintaining the right arm position and posture for writing.

Trouble forming letters can make it hard to learn spelling. That’s why many students with dysgraphia are also poor spellers, whether it is in language or in mathematics setting. They may also write very slowly, which can affect how well they can express themselves in writing and whether they can keep up the pace of work in a classroom.

Having dysgraphia does not mean a child does not have intelligence. And, when students with dysgraphia struggle with writing, they’re not being lazy. But they do need appropriate, and meaningful help and support to improve not just in handwriting, but their ‘learnability.’

These deficits can reveal themselves in different aspects of written communication and computational procedures. The comorbidity of these disorders, in the case of many children affect each other. For example, poor spelling that may be one component of dysgraphia can be the result of reading difficulties like dyslexia. I have seen many students missing points on examinations because of poor handwriting and poor organization of symbols in mathematics expressions, equations, and inequalities.

The affect of agraphia and dysgraphia on mathematics learning and communicating is multi-fold. Poor writing in mathematics combined with the lack of conceptual understanding makes it difficult to detect the nature and type of errors in the written work. For example, it is difficult to determine whether the error is just because of writing or lack of concept formation, misconception, or language. For example, even if one found the error in the writing part, one may not be able to correct it as the corresponding conceptual schema may be absent. And, poor writing, itself may contribute to many misconceptions in mathematics learning and communication. For example, for the problem:

the student may plan to write it as: 4 2/3 + 3 3/4. In this case, the student may write and read the expression as: 42/3 and 33/4 and may answer it correctly, but the error is just due to writing. Or, on the other hand, he may not know the difference between the expressions, then the error is due to lack of conceptual understanding. I have found that errors due to handwriting are relatively easier to correct.

Despite the prevalence and significant impact of developmental dysgraphia on reading, language, and mathematics, the topic has received relatively little attention from researchers, particularly, in the context of writing in mathematics situations. You cannot do much mathematics without writing.

C. Evaluation for Dysgraphia

For years, dysgraphia was an official diagnosis. It no longer is. But there is a diagnosis called specific learning disorder with impairment in written expression. This refers to trouble expressing thoughts in writing, rather than just transcription difficulties. Getting a full evaluation of dysgraphia, therefore, is the first step, so one can help a child better understand his/her challenges and strengths. Along with helping a child, it is important that we improve the teaching of handwriting in our schools so that we do not have many more students who may demonstrate symptoms of dysgraphia. But, more importantly, our instruction is such that the numbers of acquired dysgraphia disappear.

An experienced teacher knows it right away whether it is dysgraphia or acquired dysgraphia. Evaluators have ways to identify the transcription challenges, though. Some tests for writing include subtests for spelling. There are also tests for fine motor skills (the ability to make movements using the small muscles in our hands and wrists). And there are tests for motor planning skills (the ability to remember and perform steps to make a movement happen). Neurologists and neuropsychologists evaluate the condition for the neuro-psychological reasons of the condition.

Special professionals evaluate students with trouble writing. Occupational therapists and physical therapists can test motor skills. Trouble with writing can be caused by other learning challenges, too. To get the right help for a child, it is important to know what is causing a child’s difficulties. A free school evaluation can help one understand these challenges, along with the child’s strengths.

D. The Visual System and Dysgraphia/Dyslexia

Visual processing is a higher cortical function. Decoding and interpretation of retinal images occur in the brain after visual signals are transmitted from the eyes. Reading and writing involves adequate vision and the neurologic ability to identify what is seen and is written. Although vision is fundamental for reading and writing, the process is more than seeing. The brain must interpret the incoming visual images to construct meaning. Historically, many theories have implicated defects in the visual system as a cause of dyslexia, dysgraphia, and/or dyscalculia. Research has shown that many of these theories to be untrue. A series of studies have systematically demonstrated that deﬁcits in visual processes, such as visualization, visual sequencing, visual memory, visual perception, and perceptual-motor abilities, are not basic causes of reading and writing difﬁculties. Difﬁculties in maintaining proper directionality have been demonstrated to be a symptom, not a cause, of reading or writing disorders. Word reversals and skipping words, which are seen in readers with dyslexia and dysgraphia, have been shown to result from language deficits rather than visual or perceptual disorders.

Multiplicity of actions and processes are called upon to make the reading, calculation, and writing process to come to fruition. For example, short-duration, high-velocity, small jumping eye movements called saccades are used for reading. Another human ability that makes reading and any writing possible is peripheral view—our ability to view a string of symbols. Any disturbance or lack of mastery in these skills may cause difficulty. For example, readers with dyslexia characteristically have saccadic eye movements and ﬁxation ability similar to the beginning reader but show normal saccadic eye movements when content is corrected for ability. The saccadic patterns seen in readers with dyslexia seem to be the result, not the cause, of their reading disability. Decoding and comprehension failure, rather than a primary abnormality of the oculomotor control systems, is responsible for slow reading, increased duration of ﬁxations, and increased backward saccades. In mathematics, however, fast readers also have problems in mathematics as their comprehension is reduced by this process. Children with dyslexia often lose their place while reading because they struggle to decode a letter or word combination and/or because of lack of comprehension, not because of a “tracking abnormality.”

However, improvement in reading has been shown to change saccadic patterns, but there has been no evidence to suggest that saccadic training results in better reading. Finally, children with saccadic disorders do not show an increased likelihood of dyslexia. Dyslexia is not correlated with eye or eye-movement abnormalities. Other conditions may affect reading. Convergence insufﬁciency and poor accommodation, both of which are uncommon in children, can interfere with the physical act of reading but not with decoding. Thus, treatment of these disorders can make reading more comfortable and may allow reading for longer periods of time but does not directly improve decoding or comprehension. However, supervised practice in physical formation of symbols and use of graph-paper improves handwriting.

Numerous studies have shown that children with dyslexia or related learning disabilities have the same visual function and ocular health as children without such conditions. Speciﬁcally, subtle eye or visual problems, including visual perceptual disorders, refractive error, abnormal focusing, jerky eye movements, binocular dysfunction, and misaligned or crossed eyes, do not cause dyslexia or dysgraphia. In summary, research has shown that most reading and writing disabilities are not caused by altered visual function.

Many children with reading disabilities and grapho-motor dysfunctions enjoy playing video games, including handheld games, and comic books for prolonged periods. Playing video games and reading comic books requires working memory, concentration, organization, flexibility of thought, visual perception, visual processing, eye movements, and eye-hand coordination. Convergence and accommodation are also required for handheld games. Thus, if visual deficits were a major cause of reading disabilities, dysgraphia, and dyscalculia, students with such disabilities would reject this vision-intensive activity.

E. Understanding Handwriting Processes

1. Development of Handwriting

‘Writing’ can refer to the basic act of producing written letters, numbers, symbols, and words as well as the complex act of planning, organizing, writing, shaping, and proofreading a text. It is a complex process that requires the coordination of motor planning and motor execution in addition to brain processes of organization, executive function, and language ( which work together to constitute the functional writing system.

(a) Pre-School Years

In the preschool years (3 to 6 years), children learn the basic transcription skills necessary and preparation for coordinating the visual and motor systems when copying symbols of any kind. They should practice copying and drawing in free-hand. They should trace and touch three-dimensional and two-dimensional objects. They should be ask to make, draw, and copy—open objects (making lines, angles, rays, line segments, half-moon, arcs, intersecting lines, parallel lines, etc.), closed circles (making circles, big and small dots, triangles, rectangles, etc.)—using different kinds of media—on sand, water, paper, air, sky, iPad screen, etc. with pencil, chalk, crayon, pen, fingers, stylus, brush, etc. Children learn topological concepts before they learn Euclidean geometrical concepts.

Before children write numerals and letters formally, they should practice drawing vertical, horizontal, and slanted lines, circles, half circles, angles, making loops, etc. The purpose of this activity is to get them the flexibility in use of their hand and development of fine motor coordination, orientation, joining lines, shapes, recognizing corners, etc.

Children should be helped to practice forming numbers/numerals with the same diligence, care, and focus as they practice writing letters or playing sports. Along with practice with forming numbers and letters, children should also learn and master the identification of left from right, up and down, inside and outside, closer and farther, corner, turn, arrow, and teachers should refer to the movement of their hand in forming numbers with proper directions (e.g., left-right and up-down orientations).

(b) Early Childhood Years (First through Third Grade)

Typically, children begin learning to write formally in kindergarten and first grade, with continued development in second grade. In addition to learning the motor tasks required to write letters and numbers, the child must be sufficiently familiar with the language and the associations between words and sounds. Similarly, they should be sufficiently familiar with the numberness before they write numbers. By third grade, most children have established automaticity with writing, wherein the movements required to write letters have become rote response patterns. This means many of the writing habits and letter and number formation are very much set. Therefore, the first three years (K through Third grade ) of work on writing is crucial for forming good ‘handwriting habits.’ However, recent research suggests that handwriting can continue to develop and improve well into the third grade, even while automaticity is emerging.

In general, unfortunately, many teachers in the United States no longer explicitly teach the process of writing letters and numbers, which can hinder those children who struggle to master this skill. On the other hand, it might be an international trend; handwriting is not a propriety for teachers.

(c) Upper Elementary Grades

Writing tasks beyond the early school years require higher-order language and numerical processing and executive functions to organize, plan, and execute a coherent and cohesive product. Writing a sentence, for example, requires that the child internally generate the statement, segment the statement into sections for transcription, retain these statement sections in memory while writing, and check the completed written statement against the internally generated thought. Writing a paragraph or essay requires planning, organization, execution, and proofreading to ensure that the statements create a coherent argument or thought. If a child has not achieved automaticity in writing by the third grade, he or she is likely to experience greater difficulty in writing as academic expectations require cognitive processing beyond the motor aspects of writing.

On average, children spend up to half of their school day in tasks that require writing, and the development of handwriting has been correlated with academic achievement. Automaticity in letter- and number-writing is a good predictor of quality and length of written assignments in elementary, high school, and college; but impairments in any part of the writing process can interfere with a child’s ability to produce written language at an age-appropriate level.

2. Writing as a Cognitive Function

The act of writing is the demonstration of the development and functioning of different component skills of gross and fine motorcoordination and executive function (e.g., working memory, inhibition control, organization, and flexibility of thought, etc.). Executive function is the ability to engage in purposeful, self-regulated, and goal-directed behavior. Executive functions and effective writing are inseparable, even in tasks such as: formation of letters and numbers. In fact, writing is an executive function task. Executive function supports handwriting and practicing handwriting activity supports the development of executive function.

Before a physical activity is executed, it is a mental activity—it is mentally executed. One imagines the activity (in the working memory). As one envisions a physical activity, decides to engage in it, one mentally plans it execution —

Where and how to begin? (focus—inhibition control and organization, planning)

Is there enough space for the letter or number? Should I write the number to the right or left, small or big, above or below? (organization, planning, spatial orientation/space organization)

Should I write smaller or larger than what is already there? (Analysis)

Writes the number (plan is executed)

Is it the right size? (organization, evaluation, reflection)

Initially, all writing (just like any physical activity) is done consciously, deliberately, and laboriously, then it is done part consciously, and after accuracy (if properly supervised), fluency should be achieved and with practice automatization is reached. Then, the physical part of writing is executed unconsciously (kinesthetic memory, long-term memory,organization).

When the physical action of writing a number is finished, the image still lingers in the working memory. After each practice, the residue of that action, in the memory system and the muscle memory is enhanced. This process helps achieve fluency and automatization. Repeated practice of organized writing makes changes both in psycho-motoric and cognitive skills. It leaves its mark in the form of new neural connections and thicker size and more myelanization of the neurons. A neuron (dendrite-axon) starts as a narrow lane and with practice it becomes a major highway for transmission of information. The writing process affects cognition. The type, amount, and nature of writing also have differentiated affect on cognition. When writing is automatic, one can think and write and can use writing as a demonstration of learning and, then means of new learning.

Initially, forming the orthographic image of a number is an individual’s need and desire to represent quantity after a successful quantitative experience —the urge to demonstrate this new learning. We should ask children to practice writing numbers only when the child feels the need to record that quantitative experience. The need must be created. That need is created by asking: “Would you like to know how do you write what you just counted?” “let us see how this number looks like?” The child, invariably, says: “Yes!” Then, it becomes a meaningful experience.

Three to six year old children are heavily egocentric. In most cases, for a child, his name and age is very dear to him. I, therefore, always ask the child: “Do you know how many letters are in your name? Do you want to know how to write that you are 4 (or 5). Let us find out.” It is a good beginning number writing activity. Formal writing could begin with the following task.

You just counted these cubes, and you said: “There are five cubes.”

Do you want to know how to write the number five?

Look around in the room, do you see a number that is called five?

Let me show you how to write it.

Now, look at this collection of numbers written on this page (points to the number line on the board/page/wall). Can you point to me which one is 5?

If the child can correctly point to the number, the teacher/parent should ask her to trace it with her dominant finger. Then point to the diagram, on the wall/board/book that shows number 5 with the arrows for the direction and ask her to trace along the arrow on the diagram. Then, the teacher should demonstrate how to write number ‘5’ with each step clearly explained. Then the teacher should ask her to write with her finger and then by pencil on paper. The teacher should appreciate the parts of the number that are accurately written and gently correct the parts that are incorrectly written. The teacher should show the child how to correct them.

Writing numbers (and letters) depends on the nature and the purpose of the task. In the beginning, writing a number is an isolated act. The outcome is a single digit number. Later tasks will result in a multi-digit number, fraction, exponent, etc. As one can see the size, shape, and location of a number and numbers is dependent on the type of problem solving and writing task. The position, the size and the relationship with other numbers changes with the complexity, purpose and competence in mathematics concepts. Each time a new number, symbol or expression type is introduced, the teacher should clearly show how to write that kind of number or expression.

3. Issues in Writing and Remedial Instruction

Teaching writing numbers/numerals involves several skills and various physical and cognitive actions:

(a) teaching and correcting pencil grip, paper position, body position, location of the other hand, etc.;

(b) teaching and correcting poorly established letter and number construction; and

(c) teaching a new handwriting form or shaping the old handwriting form, for example, print or cursive.

It is easier to learn these skills, but difficult to correct when they are poorly taught and poorly learned. Therefore, effective initial instruction is critical. Preventive teacher is always important than corrective teaching. If preventive teacher is absent or poor then corrective teaching has to be effective and efficient. For example, after fourth grade, the items in (a) are tough to correct, but not impossible.

A correction to type (a) habit is difficult, but it can be done and it is worth it. There are definite physical reasons for this. For example, proper finger and thumb overlaps, the left-hand hook (sometimes right, but often left), and other grips are difficult to overcome and require a great deal of patience, practice, consistency, and often the use of corrective aids, such as pencil grips.

It is always a great challenge to correct error of type (b). Only if (a) is already in place, (b) and (c) are not as difficult to correct/introduce as many assume. Many immigrant children who come with proper (a) and (b) easily learn to write in English, even cursive, with little teaching.

In my private practice and in lesson demonstrations in schools, in classrooms from Kindergarten to high school, I have found many students have problems writing numbers, letters, and mathematics symbols correctly. Whenever, I find poor writing of numbers and symbols, I always help them to write them properly. I correct pencil grips, paper positions and demonstrate how to write numbers and mathematics symbols. I have even introduced cursive writing to many high school students, successfully, and eventually, they have gained automaticity.

I strongly believe we should correct writing with persuasion, with humor, with challenge, with reason, and always seeking their cooperation in this effort. I have found with effort, on my and student’s part, it is possible to correct poor handwriting. It is, of course, much easier earlier (in the early grades). However, even later, under right circumstances—when it is supported by school, parents, and other teachers, one can improve it.

I find that many students are not consistently encouraged and supported during elementary, middle, and high school to become proficient in legible and accurate writing. Writing is not an isolated activity. It is part of any academic work; it should be emphasized during all academic instruction. It is a means of learning new concepts and cognitive skills and gain new information and acquire new knowledge.

There are high school students, who have never written a complete sentence, during their entire schooling. They only fill blank spaces with words and numbers on the worksheets provided by special and regular education teachers alike. And, high school is a bit late to realize that there is something called writing.

I always make pleas to teachers/schools to have consistent handwriting instruction that focuses proper strategies and methods for writing that should include pencil grip, paper position, posture, proper use of writing equipment, and so on. Poor handwriting instruction is detrimental to learning. Both students and teachers are affected by poor handwriting instruction. For example, high school and middle school teachers pay a big price for students’ poor handwriting. It is difficult even to read an equation where the variables x, t, y, and z are involved.Mathematics expressions withfractions, exponents, trigonometric functions, radical expressions and groupings are impossible to decipher.

When I am teaching, whether one-to-one, small groups, or whole class, from Kindergarteners to graduate students, I closely observe students writing mathematics. I ask them to write a lot of mathematics. Without writing you do not develop “language containers” for mathematics concepts and the ability to communicate mathematics. Without the language containers one does not hold information in the memory. Conceptual schemas emerge from the interaction of language and models.

Writing is the recording of the abstractions and processes resulting from these interactions. Without writing them clearly, succinctly, legibly, and precisely using mathematics terminology, a student may not remember the processes and outcomes from a lesson. The writing process also helps students to connect ideas. When we do not focus on the writing process, in a mathematics lesson, problems occur.

Here are some of my observations about handwriting issues and problems from the mathematics classrooms, collected over years. These are not isolated examples. When I have observed something happening consistently, I have included it on my list.

Most students have poor grip on their pencils and their usage.

Many students, even in high school, do not know their left from right.

Most students begin writing in the middle of the page, middle of the line.

Few mathematics teachers insist students to write on graph paper.

Even when they are given graph paper, many students and teachers alike do not respect the lines on the graph paper. No instructions are given how to write on a graph paper and how s graph is an asset for writing, learning, and doing mathematics.

There is no correspondence between the numbers representing dimensions of figures and the student’s drawings. The idea of drawing a figure according to some kind of meter, scale, or unit is absent from their training.

Shapes and sizes (heights, spacing, orientation, etc.) of letters and numbers, in the same word, line, or equation are uneven. Sometimes the same variable (i.e., a) is written in the same equation, formula or expression as a, A, or looks like a 9 or q.

Most students cannot draw a decent rectangle even on a graph paper.

Writing fractions, exponents, variables and operational signs is very poor. Unfortunately, no or little instruction is given in how to write a mixed fraction or a newly introduced mathematics symbol:

(i.e, ±, <, ∏, 𝚽, &, 𝜎, @, %, ∑, etc.)

Hardly any teacher gives feedback on student handwriting or spelling with the excuse that: “I am not teaching spelling or handwriting. I am teaching mathematics.” Mathematics is an alpha-numeric language. It is not just a collection of symbols. Even in Principia Mathematica some language is used. Teaching mathematics means teaching the mathematics language: How to learn it? How to read it? How to write it? How to use it? How to use it for communicating ideas?

Most students, from Kindergarten to high school and even in my college classes, when they want to correct an error, they change the pencil to the eraser side by handing the pencil to the other hand. The other hand reverses it and hands it over to the dominant hand. The dominant hand erases it and hands the pencil back to the other hand. The other hand reverse it and hands it over to the dominant hand. One small activity becomes such a production. Some drop the pencil on the table, pick it up erase and then drop it on the table and then pick it up to do the writing. I am able to correct this problem it in one session.

And, many more ….

Students with poor grip, often have some of the sloppiest handwriting in the class or the neatest, but arrived at laboriously. Often these students write less than most of the other students in their classes. Because of poor grip, poor organization, and little practice in writing, their hand fatigues easily. A middle school student whose hand throbs when s/he writes a single paragraph is in need of effective writing instruction.

Best practices in handwriting instruction are about reducing fatigue, increasing legibility and accuracy, and achieving automaticity with comprehension. It is about activating reading/writing/spelling/concept links. It is about giving students the opportunity to communicate mathematics effectively in writing.

4. Writing Process

Writing, just like all learning, is the interaction of multiple brain systems:

• Sensory Motor and Visual Perceptual and Spatial systems

• Socio-Linguistic Systems

• Cognitive Processes and Executive Systems

• Social-Emotional Systems

It begins with the reception and comprehension of visual and auditory information. Then, one retrieves the corresponding orthographic representations to the auditory and visual information. Once an orthographic representation has been retrieved from long-term memory, assembled through sound–spelling conversion, or visual representations (information on the board, book, or paper) additional processing is required to produce an overt written response in handwritten, typed, or key-boarded form.

First, the abstract letter representations, in the working memory, must be converted to a form appropriate for the chosen output format or modality. For handwritten output, letter-form representations (e.g., a representation of lower-case print f) must be activated, whereas for oral spelling letter name representations (e.g., /εf/) are required. For other forms of output (e.g., typing) different representations (e.g., keystroke representations) would need to be computed. Here. we focus on the processes required for handwritten output only.

Some theorists assume that in generating a handwritten response, abstract letter representations are first converted to allograph (letter shape) representations corresponding to the chosen form of written output (e.g., lower-case print). The allograph representations in turn activates effector-independent graphic motor plans, which are learned representations specifying the movements (i.e., the sequence of writing strokes, the letter ‘b’ has a stick and a ball attached to the bottom right) required to write the letter in the chosen form. The graphic motor plans are effector independent in the sense that they are not tied to particular effectors (e.g., the right hand, pencil or pen, etc.), and do not specify movements with respect to specific muscles or joints. These are not innate, these are learned behaviors. They depend on earlier experiences. For example, immigrants bring the experiences of their first language in writing the English alphabets and Hindu Arabic numerals. But, the nature of writing (sloppy or neat, small or large letters, scrawls or properly form letters, etc.) are carryover in handwriting into English from their mother-tongue. They are also learned behaviors, but automatized and internalized. Hence, the graphic motor plan for upper-case print B could mediate writing of that letter with the right hand, left hand, left foot, or so forth.

Although the assumption of a progression from abstract letter identities to allographs to graphic motor plans is common, some theorists have proposed instead that abstract letter representations are mapped directly to graphic motor plans.

Regardless of how graphic motor plans are activated, the final steps in the writing process involve the conversion of the graphic motor plans to effector-specific motor programs, and the use of these plans by the motor system to execute the appropriate writing movements with the chosen effector. For example, when writing on a page, when we come to the end of the line and it is close to the end of paper, we automatically begin to write smaller or bigger depending on whether we want to finish there or go to the next line or page. There is a host of decisions being made when we write, therefore, the close relationship between executive function and writing. Ultimately, writing is a means of learning and problem solving.

During the writing process visual feedback plays a significant role, not only in ensuring appropriate orientation and spacing of letters and words across the page, but also in monitoring and controlling the shapes of individual letters. The feedback is of two kinds: first, self-monitoring/self-evaluation of one’s own work and choice. Second, a timely, supportive, constructive, and corrective feedback from a caring and knowledgeable adult. When the feedback is absent or minimal, the handwriting may become sub-standard and illegible. Most of the time, this is one of the most important contributing factor in the developing of effective handwriting.

The writing involves the integration of visual, motor, as well as cognitive and perceptive components. The perception allows one to remember and then recognize the shape of the letters and numbers that are written while sight and motor skills of the hand enable the writing. It is a continuous flow from input (visual and perception) to output (visual, tactile). Brain imaging studies show that the nerves and then a bundle of neurons are connected to these three components and then definite new neural connections are taking place or being reinforced in the writing process. In the actual act of writing, by hand with pen or pencil on paper, one must use motor skills to copy or produce a letter/number graphically, although a slower process (compared to typing), but, that motor action actually aids in a child’s cognitive development. For example, with practice, the quality and keenness of perception improves. With better perception the flow from input to output become more smooth. Perceptual improvement makes us better observers, therefore, better learners.

(a) Fine Motor Skills and Writing

Fine motor skills involved in writing letters and numbers by hand are referred to as grapho-motor skills. Fine motor skills in naming numerals by mouth are referred to as oro-motor skills. We use oro-motor skills when we speak and identify and say the number. A complex of grapho-motor skills are involved in gripping and using tools—pencil, paper, stylus, iPad, eraser, and to produce number, symbols, and letter strokes, stylus strokes, and for pressing keys when typing on keyboards. These subtle and fine motor finger and hand skills draw on executive functions: planning and control, organization,judgment, and production of motor processes in different regions of the brain.

I have observed that many middle and high school students have difficulty forming special mathematical symbols and letters in lower case and cursive form even after occupational therapy. It is understood that hand writing letters may be an important exercise to facilitate children’s early letter understanding. At that time, many occupational therapists (OT) and handwriting specialists work on the improvement of generalized motor skills rather than specific fine motor skills related to numbers and letters. The question is: What type of intervention is most effective: specific or general—whether this effect is general to any visual–motor experience or specific to handwriting letters and numbers and specific symbols and forms.

Recent research has addressed this issue of letter knowledge using key precursor academic skill measures in preschool children before and after a school-based intervention. Children practiced letter or digit (numeral) writing or only viewing letters or digits. In an intervention of six weeks, results demonstrated that the writing (letters and digits) groups improved in letter recognition—one component of letter knowledge—significantly more than the viewing groups. And, digit-writing group performed significantly better on letter recognition than the letter writing group.

These results suggest that visual–motor practice with any symbol could lead to increases in letter recognition. This result can be interpreted as suggesting that any handwriting exercise, particularly digit writing, will increase letter recognition in part because it facilitates gains in visual–motor and visual-perception coordination.

(b) Cognitive Skills and Writing

Cognitive skills such as: following directions, identification of spatial orientation/space organization, pattern recognition, comparing and contrasting—comparing shapes, assessment of work, visualizing, doing task analysis, supporting one’s work with reasoning, etc. are involved when we learn and use the components of the visuomotor tasks in forming numbers—where to start, go left or right, up or down, make small or bigger, lower or higher, full circle or half circle, vertical or horizontal, in numerator or denominator, super-script or sub-script, etc. Students who write mathematics have a larger mathematics vocabulary, remember more, and receive and perceive more information during instruction.

(c) Role of Equipment in Writing

The writing equipment also plays an important role in the writing process. Few studies have explored the implications of the change of writing devices. On a very simple level, when students write on graph paper, with sharp pencils and effective erasers, in organized fashion, under guidance, their work is much better and they express that they have done better work.

The question is: what will be the nature of change in handwriting and learning as move from traditional pen or pencil on paper to computer keyboards, digital stylus, pens, and fingers on writing tablets, and speech to print software? Results from analysis of previous literature on various writing methods, devices and their implications have shown that there is a significant difference (particularly on neural activation, formation of connections, and on the myelanization process) between handwriting and the use of mechanized devices. Neuroscientists have noted that the shift from handwriting to mechanized or technical writing has serious implications on cognition and skill development—and the whole learning process. However, there is not enough research in this area. However, we can extrapolate some of the implications from the available research.

For example, typewriting involves both hands actively while handwriting involves, one active hand and the other as an aide (holding paper, maintaining balance, holding the book or another device from which information may be copied, cleaning and straightening paper), and handwriting is slower and more laborious than typing. In handwriting, one may focus on the word as a whole—the gestalt. Handwriting needs a person to shape and organize a letter, where typing does not. Some Japanese studies have shown that repeated handwriting aids in remembering the shape of the letters and numbers better. One study showed that when children learned words and numbers by writing, they remembered them better than if they learned them by typing. There are some observational studies to show that because of the topological nature of the written information (spatial location on the page), when we read that information from the book, or write it in a particular part of the paper, we remember better. I still remember many passages from the books and their particular places on the page, I had read in high school or even earlier.

Handwriting makes a person focus on one point alone–the tip of the pen—and a particular stroke (part and component) of the letter or number and we look at it longer than we look at it when we type. Expert typists do not even look at the paper. I have asked many typists, if they remember the material they typed. Their answer is: no! The focus on the written letter or number heightens perception and visual motor integration. This helps focus on the task—an important component of executive function. However, mechanized writing makes a writer oscillate between the keypad, the monitor, and the source of the information— this involves constant shifting of attention. Handwriting is a better aid in developing the different components of the executive function, particularly, inhibition control, organization, and spatial/orientation.

(d) Role of Cursive Writing

Many schools do not teach cursive writing because they think it is no longer important or it is too difficult to teach. Today, the emphasis is on keyboarding. Yes, all students should learn keyboarding. It is necessary and it has an important role in the highly technological society. It is a means of acquiring new knowledge, new skills, a new avenue to empowerment. Through keyboarding, they develop many cognitive skills and other content skills.

However, the goal of writing instruction in the information age should be developing hybrid scribes who are adept with multiple writing methods, using multiple tools including pens, stylus, and keyboards. Keyboarding and print writing, alone, do not help students develop other particular cognitive skills that cursive writing, whether on paper or iPad, develops. On the other hand, a certain level of cursive writing is essential for mathematics as there are many variables that need to be written in cursive and in lower case.

Cursive writing is an important part of learning. It is a multi-sensory, multi-function activity; it is more than just a writing activity. The presence of dysgraphia and poor letter formation are two important reasons to address handwriting. Dysgraphia does exist. But, just like reading problems exist without dyslexia, mathematics learning problems exist without dyscalculia, similarly, there are many whose writing problems exist without dysgraphia. Without training in writing, writing cursive, organization, visual perceptual activity, many children show signs/symptoms of dysgraphia, without really having dysgraphia. It is acquired dysgraphia. These students, with poor handwriting, are not truly dysgraphic, they never learned organization and visual-perceptual skills—tracking, copying, structure, form, discipline, and task analysis of visual tasks. Students learn great deal of organization and structure through proper handwriting instruction.

Weak spellers and students whose letters and numbers look good but are painful or laborious to produce are two other groups who need instruction in cursive writing skills. It is worth the time to make interventions in handwriting, at any grade. Every year, I get at least five to ten students in this category, and I am happy to say that, with help and guidance, they all change for good. Their issues about writing of numbers vary from dysgraphia to poor handwriting because of: poor handwriting teaching, lack of feedback to the writing, processing speed, lack of organization, slow speed of letter and number formation, perfectionist attitude—compulsive erasing/correcting, obstinate behavior, poor grip, and no or limited experience in writing.

5. Visual Spatial Integration and Forming Numbers

Students struggling with visual-spatial-motor or visual-perceptual integration tasks often find it difficult to form shapes—letters, numbers, and geometric figures. Initially, most children struggle with forms to a certain degree, but some children’s letters and numbers continue to be inverted, uneven in size, shape, and orientation, and have an uneven amount of space between their words and digits as they copy and write letters and numbers. Children with lack of visual spatial integration misalign numbers, have poor or incorrect orientation of numbers, letters, figures and shapes. They may struggle in integrating the actions of writing on paper and looking for the information from the board, book, or paper. They often have difficulty in visualizing and representing the clusters of objects and their distribution (arrangements of concrete objects, patterns in clusters, icons, pictorial, abstract symbols). They have difficulty properly locating and forming numbers on number line. They have difficulty reading maps, staying on line, staying between lines on notebook paper, and understanding the organization of diagrams, calendars, and tables.

Visual-perceptual integration difficulty also has an effect on mathematics learning beyond just digit formation. For example, they have difficulty in understanding place value, understanding and writing fractions, decimal numbers, algebraic expressions, recognizing and drawing overlapping figures and interrelationships of different shapes and figures in geometry.

Many of these visual-spatial-perceptual motor difficulties can be prevented, corrected, or compensated if proper and efficient methods of forming numbers or drawing geometrical shapes and figures are used. Experiences with puzzles, playing multi-sensory games, toys involving spatial orientation and space organization skills develop visual-spatial-perceptual skills in formal ways. Activities involving coloring, sketching, and drawing exercises, making pattens on graph papers and with pattern blocks, free-hand drawings also achieve these goals. Solving puzzles and problems with instruction materials such as: Unifix, inch and centi-cubes, Cuisenaire rods, and Visual Cluster Cards as they have emphasis on color, shape, size and patterns help children develop executive functions and integration of visual spatial integration.

If provided appropriate experiences and support, children’s talking, reading, and writing develop simultaneously, and that progress in one area supports learning in the other. Similarly, counting objects, keeping scores in games and sport activities, reading numbers in various places and contexts, and recording quantitative outcomes of activities as numerical expressions in social and personal settings support each other and eventually, with some effort, transforms into numberness and numbersense. Many young children experience difficulty in writing, however, encouraging them by making writing as an enticing activity is an effective way to help them learn to read and know numbers and, then get them interested in writing letters and numbers. Teachers should use concrete activities (making and copying clusters of objects, tracing numbers with their dominant/writing finger, forming the shape of the number on the imaginary white board—sky writing), using language to help them notice visual-spatial information in the number shape, and describing the movement of the finger as it traces the number or geometrical shape. By talking about shapes, sizes, distances, proportions, and writing of numbers, and numerical expressions children begin to develop a better understanding of numbers, their forms, and their relationships.

Orthographic symbols (forms of numbers, mathematical symbols, shapes) should be explored and analyzed from their smallest components to Gestalt of the object by asking questions (How many lines here? How many lines and curves, dots, and crosses in this figure?). Asking them questions relating to orientations in the figures and objects also helps them see the figures and symbols better (Is going left to right or right to left? What is on top of the book?). Children come to Kindergarten knowing only the names of shapes (every object has a shape—a name). For them the name of the object is associated with the shape it possesses, but they may not be able to describe it. It is important to emphasize and discuss the differences, likeness, similarities, peculiarities, and uniquenesses of objects from the very beginning. Thus, a shape can be three—, two—, one—, or zero—dimensional. Later on, we help children represent these shapes into figures (A figure is a representation of a shape on a plane surface, like paper). And, when they have acquired the sense of quantity and the concept of measurement, we help them convert figures into diagrams. A diagram (dia = two, gram = information) is thus the integration of two kinds of information—spatial (figural/pictorial) and quantitative (dimensions/numerical).

Visualization and description of the steps involved in the task and the possible outcome improves a person’s actual performance on the task. One of the sure means, to improve visualization is doing a task analysis and then creating a ‘script of the action steps’ involved in the task. For example, we can help them in constructing scripts for writing a number. The child is seeing a number on the board, on a chart, or on a page with guiding arrows around it. The teacher helps determine the child’s dominant hand. The teacher, then traces the number and describes the action. Then the child performs the action: first traces the number with his finger and describes the action. Then, from distance, he writes the number in air. During the action, he describes the action as his finger moves:

“I begin by placing my finger at the dot on left of the page, thenI go down to the next line below, then I go right till I reach the dot on the line. I lift my pencil and start on the top on the second dot on right. I come down two lines. I have written number 4.”

He does this a few times to get a level of familiarity. Then the child repeats the action with pencil on paper using the script.

“I begin by placing my pencil at the dot on left of the page, thenI go down to the next line below, then I go right till I the dot on the line. I lift my pencil and start on the top on the second dot on right. I come down two lines. I have written number 4.”

The child and the teacher compare the ‘model’ of the number and child’s written number. If it is correct, ask the child to perform the action with his eyes closed and describe the action. Otherwise, practice it more.

This language input from the script increases visualization and the visual-motor output. The script can also be used to correct the outcome and improve their ability to write, draw, and organize. The act of writing is a means for integrating multi-faceted, multi-modal activities—aural, oral, movement, visualization, representation, abstractions resulting in forming number shapes, and cognitive functions.

Writing and recording their work is a tangible proof of the learning. Children take great pride in their work. We need to help them in accomplishing this to standards and with an awareness of quality.

E. Key Elements of Effective Interventions

If the child cannot write numerals properly and legibly, then he may not be able to express math knowledge in conventional symbols and forms in the classroom properly, clearly. As a result, the teacher and others may not be able to decipher what is being communicated about number concepts in quantitative symbols. If the child cannot write numerals automatically, speed of performing written math tasks will be very slow and math assignments may not be completed on time. Efficiency of problem solving in working memory may also be compromised on paper/pencil math tasks because numeral writing is not automatic.

Our schools are rich in language and its rituals. However, in the information age, we need to have numerate, literate, and socially and emotionally conscious citizens who can communicate their ideas clearly and effectively both linguistically and quantitatively. The objective of formal education is to achieve these objectives. The foundations of such a formal education build on the accomplishments of parents with children, at home. This begins in Pre-Kindergarten and Kindergarten. One of the pillars of this foundation is numeracy. The foundations of numeracy is solid number concept. However, development of number concept is a school-wide responsibility, not just the classroom teacher. No child should be leaving Kindergarten without mastering number concept: (a) numberness (e.g., recognizing a cluster of objects up to ten instantly), (b) orthographic image of numbers (e.g., recognize, say, and write the collection in the numerical form up to 10), (c) knowing the 45 sight facts (e.g., knowing the sight facts of numbers up to 10: sight facts of 7 are 1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2, 6 + 1), and, (d) knowing two-digit numbers (i.e., writing, reading, and using them).

The risk of any risk of possible learning disability and/or its behavioral manifestation can be minimized by preventive (appropriate, effective, efficient) teaching. To prevent the incidence of dysgraphia, an effective and efficient (proven) handwriting instruction program should be in place in every Kindergarten through second grade classroom in the country. Along with this there should be effective intervention program in place, in later grades. While specialists provide key intervention strategies for students with dysgraphia, teachers in the general education classroom have an important role to play in supporting these students as well as preventing and minimizing the affect of possible dysgraphia. An effective and efficient handwriting instruction program should be an integral part of the language program and mathematics teachers should know how to instruct students in proper ways of writing numbers, symbols, and letters.

Most elementary schools teachers intellectually understand the importance of handwriting on the quality and quantity of compositional writing, on conceptual understanding, and its contribution to all academic tasks. On principle, considering efficacy and motivation, the question, they face, is when to start formal writing and what form, particularly, cursive writing. In most English speaking countries, other than United States, cursive writing begins in first grade. It is used to be in US also, but now less than half of the elementary schools, in the country, teach cursive handwriting. I believe, we should begin handwriting instruction, including cursive, as early as possible and third grade is not too late to address handwriting problems. If the child is in grade K-2, we should have effective handwriting instruction first in print and then in first grade both. For dysgraphic children, although it is too early to identify this condition, we should provide more scaffolding, structure and intervention from Kindergarten.

Handwriting support and intervention is important and it is not never too late. The impact of supplemental handwriting and spelling instruction on learning to write has been examined in studies with first grade students. Even as little as about 15 to 20 hours of one-to-one intervention to improve children’s handwriting fluency, handwriting legibility, spelling accuracy, and knowledge of spelling patterns has been found to have positive impact in all of these areas. Such interventions demonstrate that explicit and supplemental handwriting and spelling instruction can play an important role in teaching writing to young children who acquire text transcription skills more slowly than their peers.

It is never too late to make handwriting interventions. However, handwriting intervention that includes explicit instruction and intensive supervised, effective and constructive feedback, and individual practice, even as late as fifth grade is beneficial in multiple ways. Researchers have found that even a short intervention of five hours with explicit instruction and intensive practice in writing cursive letters, words, and sentences, through fast-paced alphabet and copying activities is highly effective in increasing students’ handwriting fluency. Students also report strengthened self-efficacy beliefs for grammar and usage skills. Such studies have shown that handwriting interventions can effectively help students with limited handwriting skills to become fluent hand writers. And, handwriting fluency is important to support the further development of writing and thinking.

My experience is that meaningful interventions do work. I begin emphasizing improvement in writing for my students even if they come to me in the 3rd grade or later. The intervention is important, even if poor patterns of writing are too ingrained and they might show symptoms of dysgraphia. With help, most of my middle and high school students are able to improve handwriting significantly. The change affects all aspects of their learning and self-esteem.

1. Role of Occupational Therapy as Intervention

Many dysgraphics have poor motor planning and motor coordination issues. If a student is struggling with writing in a way that goes beyond what is developmentally appropriate, they should receive an evaluation by an occupational therapist for remediation. If they struggle to do academic and non-academic work because of the poor motor skills development, they need occupational therapy. Occupational therapy (OT) helps children acquire motor skills which can be improved with practice. These include tasks and skills that are part of learning and functioning well at schoolwork. Therapists work with children beginning with gross motor skills and then to improve fine motor skills and motor planning for tasks—academic and non-academic. However, the skills they need for writing and mathematics are subtle motor skills (e.g., coordination, spatial orientation/space organization, task-analysis, part-to-whole and whole-to-part relationships, etc.). These skills need constant honing and practice, particularly in mathematics—number and symbol forming. For subtle motor coordination, the understanding of the concepts is also needed.

Most aspects of dysgraphia can be remediated with occupational therapy to strengthen fine motor skills, support written expression, and speed up language processing. Early intervention for and orthographic processing, fine motor coordination, and written expression can help alleviate the difficulties that students with dysgraphia face. However, to be effective, the occupational therapy (related to handwriting) should be focussed and should include practice not just in fine motor skills and writing letters only, but also of numbers, numerical operational symbols (+,☓, −, ÷, ±, ∓, =, ∝, ∏, √, ∠, etc.) and other mathematical symbols (digits, fractions—proper, improper, and mixed, decimals, percents, exponents, and grouping and mathematics symbols appropriate to their grade level.

These therapies may be available for free at school through an Independent Education Plan (IEP).

2. Reversals and Inversions of Numbers

Reversal errors play a prominent role in theories of reading disability and also in dysgraphia. Similarly, some people think writing numbers incorrectly and reversals also play a role in learning difficulties in mathematics. Some people think that mirror writing is a symptom of dyslexia and dyscalculia. In fact, backwards writing and reversals of letters, words, and numbers are common in the early stages of writing development among dyslexic and non-dyslexic children alike. However, the reversal writing (mirror writing) of numbers and letters, to some extent, also culture specific. Researchers have found that in left-to-right writing cultures, spontaneous mirror writing of letters and digits in preliterate children appears more frequently on left-than right-facing characters. Overall results of this theory, drawn on neuropsychological evidence of mirror generalization, suggest that children resort to a right-orienting/writing rule when learning to write. These results constitute a further illustration that the manifestation of mirror writing in typically developing children is culture-bound. However, in most children it is short-lived. With instruction, it soon disappears.

In general, dyslexic children have problems in naming letters but not in copying letters. Only when naming numbers and letters are a persistent problem and continues beyond first grade, this is a matter of concern. However, I suggest that, in significantly large number of cases, it can be prevented. For this we need to properly understand the nature of the handwriting issue.

As indicated above, children are more likely to reverse letter forms that face left, such as “d” and “J,” than forms that face right, such as “b” and “c”. Researchers propose that this asymmetry reflects statistical learning: Children implicitly learn that the right-facing pattern is more typical of Latin/Roman script letters. Although children who went on to become poorer readers make more errors and more frequently in the letter and number writing task than children who went on to become better readers, they were no more likely to make reversal errors. Similarly, children make more reversals with numbers/digits facing left. Unfortunately, shapes of more digits are facing left: {2, 3, 4, 7, and 9}; facing right letters are: {0, 5, 6, and 8}; and handwritten one 1 is neutral—however, typed 1 faces left.

Reversals are also a function of lack of development or poor spatial orientation/space organization skills (having difficulty seeing part-to-whole and whole-to-part and noticing and lack of understanding of the relative orientation of objects with each other and with their parts, e.g, identifying left from right and vice-versa). It is a matter of practice (playing games and toys with spatial orientation/space organization component are very beneficial), and, with consistent practice, it is a rare child who does not succeed in this task.

When children reverse numbers or letters in writing, there are activities teachers can execute to correct the problem. First, it is important that children regularly engage in activities that focus and emphasize spatial orientation and space organization, such as: playing with toys and games that have a strong spatial orientation/space organization component and asking them their right pr left hand as requested.

I have found the interventions that aim at helping children to know left from right drastically reduce the number of reversals. When they know their left-right orientation, difference between up and down, move to left or move to right, the teacher can give better and more specific directions in forming letters, numbers, and shapes, using the orientation of the movement of the pencil in relationship to their left or right. For example, to write the number/numeral “2” “you begin by putting the pencil tip on the dot on the top and then go to right making a ‘half circle/moon facing left’by ending at the bottom of the placewhere we started and thengo right, directly below the half moon.” The teacher demonstrates this action several times with the script and invites several children to come and trace the auxiliary lines and the number by the dominant index finger.

When I see a child reversing a number in writing on his/her paper, I ask the child to come to the board and write the number on the board. If he reverses the number on the board, I ask him to identify that number somewhere in the classroom several feet away from the board. I ask the child to look at his/her number and the number identified in the classroom and then ask: Does your number look like the number you just identified? If the child answers: ‘yes!’ Then write the number in-front of him on the board next to his number. At that point, generally, the child says: ‘Mine does not look like yours.’

If the child still does not see the difference, I ask the child to trace the number written on the board and then go trace the number identified in the room. Then, I ask the child to identify the number in another place in the room, and trace the number by his dominant hand’s index finger. The correctly formed, identified number should be several feet away from him and then I ask him to come back to the board and write it. You repeat the process till the child writes it correctly

I have also used the following process for helping children to correct b/d and 2/5 reversals and other number reversals with positive results. Simply point out to the child that the letter ‘b’ (when printed correctly starting with a line) is a stick and a ball, at the bottom, to the right of the stick.

For example, to make the stick, start with a line from top to bottom and then attach a ball to the right of the line/stick’s bottom (a little later refine it to almost a circle).

On the other hand, the letter ‘d’ is a ball and a stick, and the ball is to the left of the stick in the bottom.

To make ‘d,’ start to make a ball at the ‘dot’ in the bottom and then place a stick to the right next to the ball. The ball is to the left of the ball.

I have helped many children, teens, and even adults to learn these letters correctly, in as little as five minutes. In the case of numbers, I show them the correct directions of forming the letter and repeat the above process.

I also believe that the ‘b/d’ confusion and number reversals ‘5/2’ happen when we start students learning to read and write (more properly guess, in this case) before they have mastered the proper production and identification (orally, by naming them) of all the letters of the alphabet fluently and the numbers and their clusters correctly.

3. Vision Therapy and Dysgraphia

However, there are others, because of difficulty in adjusting to change in visual focus during certain reading and writing processes, particularly reading a mathematics information (equation/formula) from the board to transferring that information to the paper on the desk, find it difficult to write and they develop distaste for reading and writing. They avoid reading and writing. Some of them even complain of headaches due to this problem in vision focus. Such students have been may benefit from vision therapy and some of these students are helped a great deal by such vision therapy. They report diminished headaches and they became better at accommodating and shifting focus between distance and their book or paper.

Many professionals advise vision therapy as a helpful if there are underlying accommodative or motor/fusion problems. Even some clinical psychologists and neuropsychologist specializing in Dysgraphia also advise for vision therapy for dysgraphics with what is essentially a subtle motor problem.

4. Role of Cursive Writing as Intervention

When I teach (initial teaching, intervention or remedial instruction) and when I observe students with poor handwriting in mathematics setting, I work with them on handwriting. If in arithmetic expressions their numbers and symbols are undecipherable, in algebraic equations and expressions, they are not able to differentiate between “+” and their written ’t.’ They read symbol ‘x’ as symbol of multiplication symbol and as variable, rather than writing ‘x” as variable and ‘⨯’ for multiplication operation, and many other similar situations. I always work with them on graph paper. More than half of the reversal, organization, and handwriting problems disappear because of the proper graph paper usage.

I also enlist the help of parents, teachers, and naturally, the occupational therapist to provide the handwriting support with numbers and arithmetic and algebraic expressions. I insist that letters in equations and expressions are written in cursive. Children who are introduced to cursive early, I have observed, make fewer reversals and inversions.

Printing is a discrete activity, whereas, cursive writing is more continuous and fluid activity. Moreover, cursive writing more accurately portrays the blending of phonemes in words. Letters flow into each other just like the sounds to form speech. In cursive writing, there are no spaces between letters just as there are no spaces between sounds. A word written in cursive is a better analog for the spoken word, both visually and motorically.

Cursive writing has fewer starting points than disconnected print letters, which translates to improved writing speed, more consistent letter sizing, and neater overall appearance of writing. Cursive writing provides more flexibility in hand movement and less hand fatigue and improves working memory.

Hand writing expert and educator, Mildred McGinnis (Central Institute for the Deaf) prefers cursive in her Association Method of teaching handwriting. Her rationale is that learning to form cursive letters with a single flow provides heightened cues of sequence and directionality (spatial orientation/space organization), as well as word-boundary clarity. She has designed the Association Method to teach associations between letters and phonemes using multi-modality (not just multi-sensory) techniques, with handwriting and recognition of cursive letters both being parts of her synthetic language intervention approach.

In reading of number words, I refrain from saying the number words by decoding (i.e., 124 as 1-2-4, it should be read as one hundred twenty-four—not one hundred and twenty-four, one-twenty-four, or a hundred twenty-four). At this point, I insist on asking the following questions: What digits make one hundred twenty-four? “1, 2 and 4.” What numbers make one hundred twenty-four? “100, 20 and 4” “100 and 24, or 120 and 4.” Therefore, children understand the idea of place value better when they can see these numbers represented clearly in iconic form: one hundred block, two 10-rods (two orange Cuisenaire rods), and the 4-rod (purple rod).

Teachers should be careful in teaching children how to write multi-digit numbers—how to write them (in the order they hear the components of the number, except the teens numbers), the space between digits (same), the heights of numbers (same), and alignment (by place value).

5. Strategies for introducing new symbols

In language, all the letters of the alphabet are practiced in the early education, in mathematics, however, new symbols and new shapes, and operations are introduced at each grade level. When these new symbols are introduced, it is important to help students learn them properly. They should practice them, initially, under supervision. For example, I ask: “please copy (in the written form, not in the printed form) the following equation and expressions”:

In this work, I also enlist the help of parents, teachers, and naturally, the occupational therapist to provide the handwriting support with numbers and arithmetic and algebraic expressions. Many occupational therapists do not include write numbers as part of the practice regimen. I insist that letters in equations and expressions (most of the time) are written in cursive. Children who are introduced to cursive early, I have observed, make fewer reversals and inversions.

F. Preparation for Teaching handwriting

As mentioned earlier, there are certain activities that can help teachers to motivate and prepare children to write numbers and mathematics symbols properly. Writing letters and numbers should commence when there is need to record them. A child should know what the quantity represented by the number 5 is before they are asked to write the orthographic image of the number 5. Writing too early before knowing this, in most children’s case, may not be appropriate, as it becomes just a vacuous activity. When children are asked to write (letters and numbers) in their journals from Day1 of kindergarten, in many cases, it may be counter productive. This is before some children even know how to correctly form letters and numbers. By this process, the incorrect letter/number formation becomes ingrained along with incorrect spellings. Later on, even with efficient instruction, it is difficult, but not impossible, to unteach what they have already mis-learned (i.e., incorrect starting of letters and numbers).

As a result, many children develop idiosyncratic ways of writing numbers, for example, from the bottom up, mixture of lower and upper case letters in the same word, uneven spaces, reversals, etc. Teachers should introduce children to the proper way of forming the numbers, on proper writing paper, and with the right equipment at the right time. Proper formation of numbers (size, spatial orientation, location, etc.), is important from the beginning so that they are aware of the standards of handwriting and are prepared to discriminate between different types of numbers and then can focus on the quality of writing.

However, they should have number work, such as counting forward and backward, organizing objects in groups, playing with dominos, dice, Visual Cluster cards, Cuisenaire rods, geometrical shapes, puzzles, and games from day one. So that writing number and number words becomes meaningful and related to quantity.

1. Identifying child’s dominant hand:

Identifying the proper writing hand is important. To do this, the teacher offers writing utensil towards center of the chest and the student writes name under teacher observation. Student switches hand and writes again. Teacher offers writing utensil towards center of the chest and the student draws a house with teacher observation. Student switches hand and draws house again. Teacher notes:

How does the student hold the pencil?

Does the student switch hands during task?

What is the quality of the writing/drawing?

How well organized is the writing/drawing?

How quickly does the student complete each task with which hand?

When a student’s preferred hand is identified, discuss ways that he or she can be helped to remember which hand to use. Remind the child about it as many times as needed. Some parents provide a watch or bracelet to wear on the writing hand or teachers can mark the hand with a sticker or a marker or mark the correct side of the desk. However, it should be for a short time. All over the world, children can identify their right hand by the fifth week of the Kindergarten, definitely at the end of Kindergarten. I have demonstrated to teachers that they can help children to learn left from right and impulse control if they follow only two instructions in the class:

(a) Please raise your right hand when you ask me a question or want to answer a question, and,

(b) When I call a child’s name, that child is the only one who answers my question, but by first raising the right hand.

Within few weeks, children learn to recognize their right hand from left hand. I do the same thing in other grades.

If a student’s hand preference is still not clear, a timed pegboard activity based on the Jansky Kindergarten Index can be helpful. The teacher places a pegboard and pegs in front of the hand. The teacher allows 30 seconds and should note the number of pegs and their arrangement, and the dexterity with which the student places the pegs. The student repeats with opposite hand.

2. Practice spatial orientation/space organization:

Skills such as recognizing left-right orientation, up-down, side-ways, circle, half circle, straight-line, vertical, horizontal, next to, dot, arrow, sketch, draw, column, row, etc., are highly correlated to the skills essential for the formation of letters and numbers. Children with left-right identification difficulties make more reversals and inversions. Spatial orientation/space organization skills are also directly related to several mathematics concepts, such as: place value, fractions, decimals, exponents, geometry, integers, etc. The child who is proficient in spatial skills, the chances of reversal and transpositions are minimal and if they happen, they disappear fairly soon with little practice.

A teacher should always be aware of the fact that the current learning activity with children is not a terminal activity. It may be an extension of skills learned earlier and may become a basis for other complex skills and concepts to come. Therefore, she should be aware of the trajectory of the concept(s) involved in the activity—how did this concept evolve and how does this activity prepare the child for future forms of this concept/task—more complex, longer, more involved, etc. For example, in Kindergarten, they may only write only single and two-digit numbers in the operations, but, later on they will see more multi-digit and more complex numbers and the writing task will be more involved. Therefore, aligning of digits, sizes of numbers, space between then, and their location and orientation is important. The following numbers have different values because of their spatial arrangements and, relative positions, and orientation, therefore, children’s early mastery of spatial orientation/space organization is very important.

23, 2^{3}_{, }2_{3, }2.3, .23,…

Mathematics demands from a child a higher level of sophistication about the skills of space organization/space orientation. And this skill changes with age and cognition. Space organization/space orientation has three stages of development:

Ego-centric stage (knowing relative position of objects using one’s own body parts; i.e., to my left, to my right) (acquired around age 5-6),

Opposite perspective (knowing the opposite perspective (to your right, to your left) (acquired around age 7-8), and,

From any perspective (to the left of door, right of school building from front, opposite to the hypotenuse, above right of 2, superscript, sub-script, etc.) (acquired around age 10-11).

For number related writing, one needs only the first stage of spatial orientation/space organization. For other concepts and procedures in mathematics: fractions, procedures (long division, solving equations, graphing, geometrical designs and proofs, etc.), responding to teacher’s questions, transition of positions in other orientation (rotating, reflecting, etc.) and one needs second and third stages of spatial orientation/space organization. For example, in geometry proofs, the drawings are complicated and one has to discern hidden figures and drawings need perspectives, in trigonometric ratios one has to consider concepts such as: opposite to right angle, ratios of different sides, etc.

3. Psycho-motoric activities and organization

To teach numeral/number formation one needs to include gross and fine-motor activities, explicit instruction in directionality and sequence of steps toward automaticity before moving to fine motor activities: shaping the number, quality of handwriting, alignment, size, space between numbers, type of writing—lower case vs upper case, vs, cursive, etc. In our approach, we use gross motor activities like sky writing along with orally describing the set of directions and sequence of letter/number formation from the beginning. We find many students in remedial settings, and even in regular classes, who write huge, uneven, ill-formed numerals with uneven distance between them. Some of their writings are not even unrecognizable. When such students (with poor handwriting and also dysgraphics) get to algebra, they cannot work with both letters and numerals efficiently to show their work. The gross motor activities that are helpful, include: finger paint, numeral formation in rice, flour, or sand, shaving cream, on glass surface, etc. Today the iPads and other Apps provide a very nice surface to practice writing, if it is properly used. Unfortunately, because of lack of standards in writing and lack of focus on quality of handwriting, these equipments are being misused.

4. Paper (writing surface) positioning and efficient pencil (stylus) grip

The teacher should sit on the same side of the table as the child and shows how does she hold the pencil in her hand. Then she should show how she picks up the pencil and writes, turns the pencil in the same hand (twirling) and erases, and then turns the pencil in the same hand (twirls) and writes again. All of this is done having the pencil in one hand (by the twirling motion). Then she gives instructions to repeat her actions: holding the pencil correctly—proper holding, proper pencil grip, right movement, proper turning of the pencil (twirling), etc. Then, she instructs:

Place your pencil on the desk with the point toward you.

Pinch your pencil with the index finger and thumb in a pinch position.

Lift your pencil.

Turn the pencil from writing to erasing and back to writing mode (twirling). Teacher should show the proper use and placements of both hands in writing. One hand to write and the other to hold and balance the paper and to balance the body. Second hand is for holding the paper, cleaning the erased space, holding the book or the iPAD while copying from it, etc.

Teachers should ensure that students lightly grasp their pencils approximately 1 inch from the point or where the point begins. As students lift their pencils, they will fall back into correct writing positions and rest on the first joint of the middle finger. They should practice as she narrates the instructions. After a few practice sessions with students, students will only need to hear the directions. For example,

“Stop, pinch, lift to adjust your pencil grip. Write, erase, clean, write, ….”

The use of a plastic pencil grip or a metal writing frame can aid students in changing a fatiguing grip to a typical, less tiring one. Pencils with soft lead require less pressure from the student, thus reducing fatigue.

Many handwriting programs recommend that when using manuscript writing, right handed students keep their papers parallel to the bottom of the desk to help them keep their manuscript letters straight. Left-handed persons should keep the edge of the paper parallel to the writing arm which should be approximately at a 45 degree angle to the edge of the desk. (Teacher should demonstrate the positions).

In cursive, right handed students should keep the right corner higher than the left, whereas the left handed should slant in opposite direction. This allows students to see what they are writing and avoids smudging as their arms move across the page. It also prevents hooking or a curled twist. In all cases, students should anchor their papers at the top with their non-dominant hand. There are many videos and pictures on the Internet for holding pencils. The teacher should share a video like that or she should prepare her own video. It is better to prepare one’s own video as children feel secure to hear their teacher giving directions.

5. Following sequential visual and oral directions

Use of visual cues (numbered arrows, dots to show starting points) to guide number formation, following scripts for tasks completion and verbalizing, visualizing the task and verbalizing or acting it, and staying within lines and squares on the graph-paper is important. I always work with my students on graph paper with proper instruction about how to use the graph paper and how to write the newly introduced symbols accompanied with continuous feedback. Mathematics work should always be done using graph paper for classroom work, math assignments, and tests.

In the introduction of graph paper to children, as early as Kindergarten, I talk about words, we are going to use: columns, rows, squares, cells, corners, straight lines, vertical lines, horizontal lines, top-to-bottom, sideways, etc. My refrain during teaching mathematics is “respect the graph paper and the symbol you are writing.”

Graph paper helps students with dysgraphia stay within the lines, which becomes increasingly important in the later grades when they are faced with more complex math tasks. It reduces unintended errors by providing graph paper for student tests or as a background for homework assignments. The size of the graph paper should be according to the grade level. In earlier grades (K and First-grade), the graph paper may have 1 centimeter by 1 centimeter squares (cells), and they may write one digit in each square. In later grades, it should be the standard size. In the initial learning of number formation, verbalization of consistent, precise directions for forming each number is necessary. Teachers should teach naming of numerals and orally starting steps of ordered procedures for forming numerals. Using graph paper for mathematics work minimizes the errors related to (a) mis-alignment of digits, (b) the orientation and aligning of multi-digit numbers, (c) the location of exponents (super-scripts), sub-scripts, fraction notation, etc.

Structure, organization, adhering to protocols, and proper and immediate feedback is the key for improvement not only in mathematics learning, but also in improving handwriting. Half of the reversal, organization, and handwriting problems disappear because of proper graph paper usage.

G. Formal Teaching of Writing Numerals

Much of what young children understand and learn about their world comes to them through their senses as they interact with others—objects and people. Through the tactile and kinesthetic sensory input, they learn concepts about shapes and objects:

their contours—extent, corners or no corners,

their range—how big, how far, how close, how small;

quantities—their organization, their spread and relational locations—how many and how much;

objects—their similarities and differences, their locations, their positions, their relationships with each other.

From these interactions emerge the need of knowing concepts such as distance (i.e., length, height, width, etc.), spread, weight, movement and time and the urge to measure. Writing numbers, knowing numbers and their relationships, and understanding various roles of number are the means of knowing their world.

During these interactions, children notice and learn relationships such as the space between people in a line and even to realize whether it is a line or not. They observe the number, the arrangement and organization of the furniture in a room. They begin to discern letters or sounds in a word, order of letters in words on a page in a book, or order of numbers written on a package or a T-shirt.

1. Task analysis

The teacher needs to determine, select, organize, and sequence the activities in order to execute a task. For example, understanding and drawing the shape of a number is just like understanding and executing the strokes in writing a letter or hitting a ball. This task analysis and the related demonstrated execution of these steps can provide the basis for visualization of the action steps for writing the number or letter. This is not possible, if the student cannot visualize the image of the letter or number in the mind’s eye—the working memory space—the sketch-pad of the brain.

Many special needs students often have deficits in directionality, spatial orientation/space organization, and the process of visualization (lack of experiences). Many of them begin all numerals at the base line (poor teaching). They may not follow the same directionality we use for letter forms (poor or lack of teaching). They need special attention providing directions with special signs and markers and how to visualize by breaking the problem into parts (pedagogical intervention).

As mentioned earlier, the digits 2, 3, 4, 5, and 7 are written with left to right orientation, whereas, the digits 0, 6, 8, and 9 are written from right to left. There is additional difficulty with fluency for writing numerals in math in that—compared to written language which is left to right, mathematics writing work may go clockwise, counter clockwise, left to right (e.g., writing multi-digit number writing, long-division), from right to left (e.g., most multi-digit procedures—addition, subtraction, multiplication, etc.), top to bottom (e.g., fractions), bottom to top (e.g., exponents), or mixed (e.g., order of operations—GEMDAS).

The written digits and symbols can be in any place/position in relation to other digits in computations and procedures. This complicates the task and makes fluency that much more important. All of this requires a great deal of supervised practice in mathematics concepts learning and writing. We need to build automaticity through repeated practice of the sequence, directionality, and visualization for each numeral from the very beginning.

2. Flexibility and diversity of tasks:

The teacher or parent, in the writing plan, should include practice writing numbers in multiple ways: tracing the written number—first gross-motor formation of number, sky-writing the number, tracing the number by putting a rice paper sheet on the number, fine-motor—copying on paper, writing it on an imaginary board, writing with eyes closed, visualizing the shape of the number and describing its strokes, and writing it on paper. This cycle—concrete to pictorial, visualization, writing (CPVA) is not only important for learning mathematics, but also writing the number on paper. This process needs to be repeated many times.

Writing Numerals/Numbers

Task sequence 1:

No more than 5-10 minutes should be spent on the writing activity daily until it is mastered. Start with teaching numerals 1-9. The initial focus should be on observation of numeral formation—verbal description and finger movement on the shape of numerals. The subsequent focus should be on practicing on automatization of numeral writing. After, the automatization of writing numerals 1 to 9 are learned and the concept of zero is understood, then, the writing of 0 and 10 should be done.

In India, in my family, each child used to be introduced to writing in a very elaborate ritual.

A plate with a thin layer of honey was placed in-front of the child (age 3 through to 5 years), a plank of wood (called shining Patti—painted brown) with the name of the child written on top and then the letters of the alphabet below the name were placed against a platform slightly above the honey plate.

The child would be asked to look at his name. The name would be pointed by and read slowly by the patriarch of the family—in my case my grand-father (Babaji) and the child would be asked to utter his name slowly. Babaji would point to each letter from the alphabet. Then the child would be asked to trace the name on the Patti. After few times, when the child has correctly traced it. He would be asked to write his name on the plate with honey. After completing the writing the name the child would lick his finger.

It was such a “sweet” and “pleasurable” memorable experience that even today at the age of 77 years, I fondly remember the ritual and sense of the expereince. Every young child is as always eager to write. Psychologically, it responds to the egocentric needs of the child, pedagogically and socially, it meets the needs of society—transmission of culture and continuation of traditions, a memorable experience for the child, and aesthetically, it is such a beautiful occasion.

Task sequence 2:

Formally,first, the teacher should give children models with numbered arrows that show how to form the numeral in terms of which strokes to make and in what order to make them. All numeral formations begin from the top and proceed downward. Teacher traces the numeral, asks the child to study the model numeral, describe it orally, and then use the numbered arrows in the model numeral as a plan for writing the numeral.

Teacher should be sure to name the numeral as often as possible as these directions are given. The child should compare what he or she wrote with the model numeral. If the child has difficulty following the numbered arrow cues, it may help to use the eraser end of a pencil to trace over the model. In the process of giving directions, proper mathematics language—words, phrases, and terms should be used.

For example, please begin with your pencil’s tip on the dot on top, go left, then go straight down to the first line, below the dot. Then take a sharp turn at this corner to the right and make a half-moon to the right. Make sure the half-moon opens to the left. The whole time the child should have the model in front of her.

The following diagrams describe the process of forming the number “4.” The big dot on the auxiliary lines around the number 4 is the starting point (child should put the pencil point on the dot) and the arrows show the direction for forming the number (then follow the arrows). It is useful to identify the steps by placing the sequence of steps next to the auxiliary lines. This should be, first, a demonstration, then a supervised activity (few times) and finally, they do it independently. The activity can begin as a whole class activity. Once the children are able to practice writing numbers correctly and independently, we should focus on helping them write mathematical symbols, including numbers precisely and efficiently. Initially, just like staves in music, use three parallel lines on the graph paper as the guiding boundaries.

Children should practice writing only ‘written letters,’ not ‘printed letters’ (4—is a printed numeral and 4, is a writing numeral. Similarly, 1 and 9 are printed numerals, 1 and 9 are written numerals. Many children try to copy the printed numerals—with all the extra elements (e.g., the beak in the numeral 1, the bottom curve in the numeral 9, etc). This creates problems for many children later on, for example, they cannot differentiate between 1 and 7.

Task sequence 3:

Move to automatic writing after the child can form each of the 10 numerals, simple numerals are practiced first, in the order (1, 7, 9, 2, 3, 4, 5, 6, 8, 0, and 10). In writing 10, the digits 1 and 0 should be the same size, same height and the same line as the base of the two digits.

As children grow and accumulate experience, they develop the ability to form iconic and representational visual memories of their surroundings and object arrangements. They also begin to develop a sense of what their body parts and muscles need to do to turn the cap on the toothpaste tube or make a pencil move across a piece of notebook paper to form a shape or number. They even begin to organize how to hold the crayon, pencil, or pen to form these shapes, numbers and letters. Children want to share these experiences. Seeing others engaged in writing, drawing, making things, they express interest in formally recording these experiences, in drawings, making things, and writing on paper.

In the information age, we need to have numerate, literate, and socially and emotionally conscious citizens. The objective of formal education is to achieve these three objectives. The foundations of such a formal education builds on the accomplishments of parents with children, at home, it begins in Pre-Kindergarten and Kindergarten. Our schools are already language and its rituals-rich environments. The other pillar of such a foundation is numeracy. The foundation of numeracy is solid number concept and its representation—conceptually and in writing. Just like language, development of number concept is a school-wide responsibility.

3. Practice and Virtual Practice to Achieve Automatization

To achieve fluency and automatization, practice is important. Fluency should be the aim only after accuracy and precision in task has been achieved using efficient and effective methods. Student should also know the standards of performance. The question is: What should be the nature of practice. The following describes the process of achieving fluency

After a student has performed the task of writing a number correctly few times, the teacher should give feedback and review the steps the student used and connect these steps to achieving this success. This develops metacognition and confidence in the students. Then, the student should practice writing the number, symbol, or letter in three ways: (a) observe the teacher executing writing number or symbol or watch someone, preferably the teacher, performing the task on the video, (b) practice the task by visualizing it using the script for the task, and, (c) actually perform the task using the script.

Researchers have found that kinesthetic ability — which is an individual’s ability to feel an action without actually performing it, may improve an individual’s performance in the physical activity—like handwriting. The research results indicate that mental practice, i.e., the combination of action observation and motor imagery, may enhance the physical activity, they envision. Researchers claim that the same neural activity takes place and connections are made as if the person is actually performing the task.

Although, this practice helps all individuals, however, individuals who already had a good ‘feel’ for the action—that means if they have already performed it correctly, benefit the most from this mental practice (visualization of the activity). The visualization includes observing a motor imagery and listening to a script consisting of short sentences describing key visual and kinesthetic feelings associated with performing the task.

Having completed the visualization though the script, the individuals were found to have better kinesthetic imagery (KI). However, the research suggests that superior ability individuals benefit more from the mental practice intervention than those with poorer KI ability. The possible reason for this difference is that these individuals may be better at task-analysis and better observers. The findings suggest that simply viewing a video of the action or action being performed by another person may bolster one’s ability to imagine and subsequently perform that action.

H. Creating a Dysgraphia-Friendly Classroom

There are a number of strategies and decisions teachers, parents, and administrators can execute that can help students with dysgraphia challenges that are undermining their academic progress. These include supports and services at school, therapies outside of school, strategies one can try at home, and pedagogical decisions that the teacher can make.

Here are some common types of help, teachers and schools provide for students with dysgraphia.

Students with dysgraphia may get help at school through an IEP or a 504 plan. To make it happen, there are a number of accommodations and supports that are available to children, in our schools. This provides provisions for the deficits (remedial instruction) and support (better, more effective strategies) to be successful in academic programs. This includes assistive technology and tools to make them successful in academics, including mathematics. These can range from simple pencil grips to dictation software to one-to-one focused interventions. However, in any intervention, the goal should be to help students improve their content knowledge and at the same time improve their learnability. Just providing these supports without improving their learnability may not help children realize their potential.

Here are some ways teachers can make all aspects of writing easier.

ImprovingCognitive Skills

In a 2007 study, Crouch and Jakubecy studied the impact of two techniques: drill activities and fine motor activities, they found that when these activities were applied alone the results were inconclusive on which technique worked better. However, when the combination of both techniques were applied, the subject’s handwriting improved and increased his score by 50%. Therefore, this study suggests that using combination of both techniques can help improve the problems associated with dysgraphia, especially in the area of handwriting. The study shows us some direction: It is not isolated actions that improve student achievement. It is a combination of decisions and strategies that work for most students.

Teacher should have students set personal goals to improve their handwriting as they set other academic goals. Without shared goals, there is not enough motivation to persist. And, as the progress toward these goals is made, it should be publicly celebrated, in the classroom.

Teachers in grades 1-2 should allot definite time for hand writing instruction everyday. Extra handwriting instruction should be available to students who experience difficulty, even in later grades.

Children should use self-instruction/verbalization skills (teacher should help develop scripts and guided instructions for students to use) while writing numbers and doing mathematics (providing self-cues and oral scripts during the writing process—for example, “I start from the dot on the top and then I move right,” “When I write numbers vertically, I should align numbers by place value, etc.),

Encourage students to evaluate and correct number production and written mathematics during instruction and practice: (“Does your number look just like the one on the board?” “Let us check it together piece-by-piece).’’

Reinforce successful efforts at number production by appropriate praise: “your half-moon really is nice.” “What a beautiful straight line!” “May I share your number 5, under the document camera, with the group, it is so good?” “Would you like to describe it to the class the way you did it?” “Look at this graph, it is so informative and instructive!”

Give corrective feedback: “Your line is slightly higher than the other digits.” “Fractions should be written at the same level, your second fraction is lower than the first fraction.” “Would you like to make it again or just erase the extra part?”

Provide multiple opportunities and flexibility of action to write (enhance accuracy, first, and then develop fluency): “Let us write the same number in five different size pencils (colors).” “Can You design writing a number contest for the class?” “You make the rules for judging and the standards for performance for the contest.”

Teach for transfer of writing numerals to other aspects of number work—“What number comes before 7? What number comes after 7?” Then, proceed to multi-digit numerals like 27 and 605 (begin with house number, date, date of birth, etc.). “What is to the left of 7 in 27?” “What is to the right of 0 in 605?” Etc.

Classroom Materials and Routines

Teachers should provide:

Pencil grips or different types of pens or pencils to see what works best for the student.

Typed copies of classroom notes or lesson outlines to help the student take notes in the package.

Handouts for study so there’s less to copy from the board, but they must write every step, equation, unit, and words used to explain in homework and tests.

Extra time to take notes and copy material. Have two mathematics word walls (One cumulative for the year and the other topical) so they can copy the spellings, words, and symbols of mathematics during class work.

Graph paper (or lined paper to be used sideways) to help line up math problems. They must respect the graph paper—lines and cells. Students should use rulers for drawing and constructing. Frequent use of erasers.

Strategies and materials to address needs of left-hand writers.

Paper assignments with name, date, title, etc., already filled in.

Proper, accurate, and clear instructions for tasks. Make sure, students have understood the instruction before they attempt the problems.

Information needed to start writing and project assignments early. Help the students break project assignments into steps.

Rubrics and explanations to show how each step is graded.

And, in the case of unique cases,

Teacher may allow the student to use an audio recorder or a laptop in class, if they submit to you the notes as an evidence that they are using it for that purpose.

Give examples of finished project assignments as models and demonstrations.

Initially, offer alternatives to written responses, like giving an oral response to the teacher. However, finally they should submit the written assignment.

Assistive Technology

Assistive technology accommodations can support students with dysgraphia in their classroom writing tasks in all grades.

Teachers should:

Teach Typing: Typing, in particular assignments, can be easier than writing once students are fluent with keyboards. Teach typing skills starting as early as kindergarten and allow for ample typing practice in the classroom. Many schools and districts have typing programs to support their students’ budding typing skills. However, it is one of the means of expressing one’s ideas. It should not be the sole writing activity. Students should know both: handwriting and typing.

Provide Access to Speech-to-Text Tools: In very unique cases, teacher may allow this facility. There are several free, easy-to-use speech-to-text dictation tools.

Allow for Note-Taking Accommodations: Students should be encouraged to take notes. Writing is a multi-sensory activity. Copying notes from a whiteboard can be a particular challenge for students with dysgraphia. Many smart boards have the capability to print classroom notes. Allow students to take pictures of lecture notes to review later. Students can use a free optical character reader (OCR) to automatically read text from the photos to review those notes.

Be deliberate about hanging students’ written work: Classroom writing is difficult for students with dysgraphia. Written assignments may take two or three times as long to complete, and often they will look sloppy even if the student has tried to produce their best work. When displaying student work, consider waiting to hang assignments until everyone has finished so students with dysgraphia don’t feel shame at having everyone else’s work on display while theirs is still in progress. Another option is to hang typed work so students with dysgraphia can be proud of the finished product that hangs alongside the work of their peers.

Completing Tests and Assignments

And, in the unique cases,

Adapt a portion of test formats to cut down on handwriting. For example, use “circle the answer” or “fill in the blank” questions.

Use a scribe or speech-to-text so the student can dictate test answers and writing assignments.

Let the student choose either print or use cursive for handwritten responses. But, they must practice both.

Allow a friend/classmate to “proofread” to look for ‘handwriting’ errors, not the conceptual and procedure errors.

Provide extended time on tests. I have always provided the provision of taking the tests as many times as they wish. When students do better, the number of re-takes decrease drastically.

Provide a quiet room for tests if needed.

These are just a few things that may help a student with dysgraphia in the classroom. But mainly it is important to be creative about accommodations and to communicate with students about their individual needs. When students feel understood and supported in the classroom, the positive social-emotional impact has ripple effects through their whole school experience.

Bibliography

Andrews, J. & Lombardino, L. (2014). Strategies for teaching handwriting to children with writing disabilities. ASHA SIG1 Perspectives on Language Learning Education. 21:114-126.

Berninger, V. W. & Wolf, B. (2009). Teaching students with dyslexia and dysgraphia lessons from teaching and science. Baltimore, MD: Paul H. Brookes Publishing.

Berninger, V., & Wolf, B. (2016). Dyslexia, Dysgraphia, OWL LD, and Dyscalculia: Lessons from Science and Teaching (2nd ed.) Baltimore, MD: Paul H Brookes Publishing.

Crouch, A. L., & Jakubecy, J. J. (2007). Dysgraphia: How it affects a student’s performance and what can be done about it. Teaching Exceptional Children Plus, 3(3) Article 5. Retrieved [11/30/2019] from http://escholarship.bc.edu/education/tecplus/vol3/iss3/art5

Csikszentmihalyi, Mihaly (1990). Flow: the psychology of optimal experience (1st ed.). New York: Harper & Row. ISBN9780060162535.

Fuchs, L.S., Fuchs, D., & Malone, A.S. (2017). The Taxonomy of Intervention Intensity. Teaching Exceptional Children, 50 (1), 35-43.

Moats, L. C., & Dakin, K. E. (2008). Basic facts about dyslexia and other reading problems. Baltimore, MD: The International Dyslexia Association.

Shaywitz, S.E. (2003). Overcoming dyslexia: A new and complete science-based program for reading problems at any level. New York, NY: Alfred A. Knopf.

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 Common Core State Standards in Mathematics (CCSS-M)— set for arithmetic for early elementary grades are specific. To have mastery in a particular concept or procedure students should have its:

(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:

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—number concept, place-value, fractions, integers, idea of variability, and spatial sense. These problems may manifested in cognitive delay, neurological deficits, lack of preparation in pre-requisite skills for mathematics learning, or lack of experience related to number, quantity and space. The locus of developmental learning difficulties is in the learner. It may either be inherited from a parent – genetic, or the result of a combination of both parents’ genes – congenital and poor developmental patterns.

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)required for an operation. When a child has not mastered the number facts, he cannot benefit from this stored “verbalizable information about numbers” that is used with prior associations to solve problems involving addition, subtraction, multiplication, division, and square roots.

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:

Memory problems (e.g., short-term–reception, working–manipulation, and long-term memory—retention);

Inhibition control (difficulty maintaining concentration, and focusing on the appropriate concept, procedure, or skill),

Organization (visual-spatial confusion, lack of organization in physical space, working equipment, or ideas, skills, and working scripts, etc.),

Flexibility of thought (rigidity in using only a limited strategy—for example, using counting as the only means to derive facts)

Information processing difficulties (e.g., the cumulative nature of mathematics calls for heavy demands on processing information), and,

Motor disabilities–dysgraphia (graphomotoric and pscho-motoric issues—poor drawings, writings, lack of clarity in executing procedures).

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 visualization. To be successful in mathematics, one needs to be able to visualize numbers and mathematics models and situations. Students with dyscalculia have a difficult time visualizing numberness and estimation and often mentally mix up the numbers, resulting in errors and misconceptions.

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.

1. What is the incidence of dyscalculia?

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.

3. Is a dyslexic individual likely to be dyscalculic?

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.

4. Is dyscalculia widely understood?

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 withdifficulty 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 helping students construct. The second component is very important. It means, helping them learn ways to derive and learn efficient strategies that help them in deriving facts. It also means helping them become better learners. This means learning and memorization techniques for arithmetic facts, making connections, seeing patterns, acquiring study skills and metacognition—learners identify strategies that help them to learn. These are important principles in instruction for all children, but particularly for students with learning difficulties in mathematics. It is adherence to the six levels of learning mathematics:

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 constructingEmpty 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:

Sequencing and task-analysis (breaking down the task into parts and then synthesizing the parts into a whole, providing step-by-step prompts),

Repetition and practice (automatizing arithmetic facts, daily testing, sequenced review)

Socratic questioning and responses (structured questioning where teacher asks process or content questions to scaffold learning and develops scripts for tasks and steps under guidance)

Control of task difficulty (the teacher provides necessary assistance or tasks sequenced from easy to difficult)

Proper use of technology (after mastering facts, concepts, and estimation)

Teacher-modeled problem solving before collaboration activities and individual practice

Strategy cues (reminders and scripts to be used in strategies)

Making connections (each concept and procedure after it has been learnt should be connected to other related concepts, procedures, and skills), and

Frequent practice and assessment of tasks to improve performance and self-esteem.

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, conceptual and procedural should be carefully included in all mathematical instruction. Before the student performs any arithmetic operation he should be asked about the language, procedure, and the conceptual model of the problem first. When he has completed the problem, he should be asked whether he knows any other problems that are similar to the one he just completed.

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: (4x=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

• Cognitive preparation

• Mathematics learning personality

• Prerequisite skills

• 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.

(b) Questioning Process

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.

4. Prerequisite Skills for Mathematics Learning

Following Sequential Directions

Spatial orientation/space organization

Pattern recognition and its extension

Visualization

Estimation

Deductive reasoning

Inductive reasoning

Teacher Characteristics and Teaching Methods

1. Mastery of Mathematics Content (n ± 3 grades)

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

a) Socratic Questioning

b) Didactic

c) Coaching

3. Empathy with the Learner and the Content

4. Teaching models

~ Appropriate to the learner and the content

~ Universal across concepts

• Exact

• Efficient

• Elegant

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

❒ ❒ ❒ ❒ ❒ ❒ ❒

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:

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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Ardilla, A., & Rosselli, M. (2002). Acalculia and dyscalculia. Neuropsychology Review, 12(4), 179-231.

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Brainerd, C. J. (1979). The origins of the number concept. New York, NY: Praeger.

Carruthers, E., & Worthington, M. (2005). Children’s mathematics: Making marks, making meaning. London, UK: Sage.

Clements, D.H. (1999). Subitizing: What Is It? Why Teach It? Teaching Children Mathematics (December), Reston, VA: NCTM

Clements, D. H., & Callahan, L. G. (1986). “Cards: A Good Deal to Offer.” Arithmetic Teacher 34(1986): 14-17. Cohn, R. (1968). Developmental dyscalculia. Pediatric Clinics of North America, 75(3), 651-668.

Copeland, R. (1974). How children learn mathematics: Teaching implications of Piaget’s research. New York, NY: Macmillan.

Dehaene, S. (1997). The number sense: How the mind creates mathematics. New York, NY: Oxford University Press.

Dehaene, S., Dehaene-Lambertz, G., & Cohen, L. (1998). Abstract representations of numbers in the animal and human brain. Trends in Neuroscience, 21, 355-361.

Dehaene, S., Spelke, E., Pinel, P., Stanescu, R., & Tsivkin, S. (1999). Sources of mathematical thinking: Behavioral and brain-imaging evidence. Science, 284(5416), 970-973.

DfEE (1999). The National Numeracy Strategy. London, UK: Department of Education and Employment.

Dunlap, W., & Brennen, A. (1982). Blueprint for the diagnosis of difficulties with cardinality. Journal of Learning Disabilities, 14(1), 12-14.

Dunlap, W., & Hynde, J. (1981). The effects of grouping patterns on the perception of number. Focus on Learning Problems in Mathematics, 3(4), 13-18.

Fleischner, F., Garnett, K., & Shepard, M. (1982). Proficiency in arithmetic basic fact computation by learning disabled and nondisabled children. Focus on Learning Problems in Mathematics,4(2), 47-55.

Fuson, K. (1980). The counting word sequence as a representation tool. In R. Karplus (Ed.), Proceedings of the Fourth International Conference for the Psychology of Mathematics Education. Berkeley, CA: University of California.

Fuson, K., Richards, J., & Briars, D. (1982). The acquisition and elaboration of the number word sequence. In C. Brainerd (Ed.), Children’s logical and mathematical cognition. New York, NY: Springer Verlag.

Geary. D. (1993). Mathematical disabilities: Cognitive, neuropsychological, and genetic components. Psychological Bulletin, 114, 345-362.

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Suggested Reading:

Chomsky, N. (2006). Language and mind. Cambridge, UK: Cambridge University Press.

Sharma, M. (2008). How to master arithmetic facts easily and effectively. Framingham, MA: CT/LM.

Sharma, M. (2008). Mathematics number games. Framingham, MA: CT/LM.

Sharma, M. (1981). Prerequisite and support skills for mathematics learning. The Math Notebook,2(1).

Sharma, M. (1981). Visual clustering and number conceptualization. The Math Notebook, 2(10).

Sharma, M., & Loveless, E. (Eds.) (1986). Developmental dyscalculia. Focus on Learning Problems in Mathematics, 8(3 & 4). Framingham, MA: CT/LM

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CENTERFOR TEACHING/LEARNINGOF MATHEMATICS

LIST OF PUBLICATIONS 2019

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 Publications Dyslexia 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

eBooks: 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

The Math Notebook (TMN)

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 on Learning Problems in Mathematics

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 Professor Mahesh Sharma is the founder and President of the Center for Teaching/Learning of Mathematics, Inc. of Framingham, Massachusetts and Berkshire Mathematics in England. Berkshire 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.com Center for Teaching/Learning of Mathematics 754 Old Connecticut Path Framingham, MA 01701 mahesh@mathematicsforall.org http://www.mathematicsforall.org

Mathematics Education Professional Development Workshop Series Framingham State University with Professor Mahesh Sharma Academic Year 2019-2020 Several 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 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 (Part III): 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 Series How 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 Subtraction For grade 3 through grade 9 teachers February 28, 2020 According 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. Algebra Arithmetic to Algebra: How to Develop Algebraic Thinking For grade 4 through grade 9 teachers March 20, 2020 According 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 Topics

Mathematics as a Second Language: Role of Language in Conceptualization and in Problem Solving For K through grade 12 teachers April 3, 2020 Mathematics 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, 2020 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.

Standards of Mathematics Practice: Implementing Common Core State Standards in Mathematics For K through grade 11 teachers (regular and special educators) June 12, 2020 CCSS-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 1220 Continuing Education Department Framingham State University Framingham, MA 01701

How do children develop the concept of number? What are the most effective ways of teaching the number concept? What is the role of prerequisite skills in learning number concepts? What is dyscalculia? This video introduces some of the strategies to answer these difficult questions.

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 University with Professor Mahesh Sharma Academic Year 2019-2020

Several 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.

A. Creating A Dyscalculia Friendly Classroom:

Learning Problems in Mathematics (including math anxiety) For special education, classroom teachers, interventionists, parents, 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 Series

Additive 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 Series How to Teach Fractions Effectively (Part I): Concept and Multiplication and Division January 24, 2020 For grade 3 through grade 9 teachers and special educators According 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 Subtraction For grade 3 through grade 9 teachers February 28, 2020 According 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 Thinking For grade 4 through grade 9 teachers March 20, 2020 According 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 Topics Mathematics as a Second Language: Role of Language in Conceptualization and in Problem Solving For K through grade 12 teachers April 3, 2020 Mathematics 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, 2020 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.

Standards of Mathematics Practice: Implementing Common Core State Standards in Mathematics For K through grade 11 teachers (regular, special educators, and administrators) June 12, 2020 CCSS-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:

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.

For many years, researchers have been exploring the mystery why certain people struggle greatly with mathematics even though they are motivated to learn it and have access to learning resources. Many people, because of these struggles, have given up on learning mathematics—even the simplest of concepts, procedures and skills. The research by cognitive scientists has focused on whether mathematics learning disability (MLD) could be due to a brain region or function that has developed differently in people who struggle with mathematics.

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.) toqualitative 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:

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 equality. Later, we can use this instructional material to derive the properties of equality, and deriving the procedure for solving equations. The instructional materials that are very efficient in developing the concept of equality are: Visual cluster cards, Cuisenaire rods, and the Invicta balance.

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 knowingeasier 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) N + N (doubles), (iii) N + N + 1 (doubles plus 1), (iv) N + N – 1 (doubles minus1), (v) (N – 1) + (N + 1) (numbers that are 2 apart), (vi) Remaining facts can be derived using these strategies. At the same time, these strategies develop and use the major elements of executive function: (i) working memory, (ii) inhibition control, (iii) organization, and most importantly, (iv) flexibility of thought.

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.

Most elementary teachers are experts in teaching cildren their native language and help them acquire reading skills–fluency and comprehension. However, many of them feel, unnecessarily, inadequate and incompetent when it comes to teaching mathematics to same children. I believe that the same kinds of skills of teaching reading and language skills are applicable to teaching numeracy skills to elementary children. There are parallels of learning to read and learning numeracy skills. In the following video, I show these parallels.

It is a beautiful article by Robert Slavin. A story like this shows why we are in the teaching profession. These stories should be multiplied manyfold so that there is no child who is left without being literate or numerate. It fills every teacher’s heart, with pleasure, when their students from their classes, courses, and workshops come and share their successes in these kinds of stories. Every teacher should read this article about Patrick’s story by Robert Slavin. Robert Slavin has been contributing to the field of education for several decades. Please share your success stories in teaching mathematics. I would love to hear them and share the story and your strategies for achieving success in yur classroom.

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 number concept^{[1]}, additive reasoning, and place value, the next important developmental concept in mathematics is multiplicative reasoning. Multiplicative reasoning is an example of quantitative thinking that recognizes and uses repetition of groups to understand the underlying pattern and structure of our number system. Multiplicative reasoning is the key concept^{[2]}in the mathematics curriculum and instruction in grades 3-4. Multiplication and division are generalizations and abstractions of addition and subtraction, respectively, and contribute to the understanding of place value, and, in turn, its understanding is aided by mastering place value. It helps students to see further relationships between different types and categories of numbers and it helps in the understanding the number itself.

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 mathematics curriculum and instruction, particularly in quantitative domain, for K through grade 4, is to master numeracy. Numeracy means: A child’s ability and facility in executing, standard and non-standard, arithmetic procedures (addition, subtraction, multiplication, and division), correctly, consistently and fluently with understanding in order to apply them problem solving in mathematics, other disciplines, and real-life situations. To achieve this: children by the end of fourth grade, should master multiplicative reasoning. They should master multiplication concept, facts, and procedure by the end of third grade and by the end of fourth grade, they should master concept of division, division facts, and division procedure. Mastering multiplicative reasoning means mastering multiplication and division and understanding that multiplication and division are inverse operations. They should be able to convert a multiplication problem into a division problem and vice-versa.

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 what it means to master multiplication tables and how to achieve this mastery. Mathematics educators, teachers, and parents have formed opposing camps about it. One group believes in achieving understanding of the concept and believe that fluency will be reached with usage, whereas the other group believes in memorizing the tables and insist that conceptual understanding will come with use. Both of these extreme approaches are inadequate for mastering mutiplication tables for all children. Both work for some children, but not for all.

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 conceptualschema 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; repeated addition, groups of, an array, and area of a rectangle.

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 learn the tables and multiplication procedure in mathematics curriculum as mere procedures--a collection of sequential steps, sometimes the facts are derived just with the help of mnemonic devices, songs, and rote memorization as ‘a job to be done.’ This means: give a cursory definition of the term (e.g., multiplication is easy way of doing addition), give the procedure (e.g., this is how you do/find it), practice the procedure (do these problems now), and then apply the procedure (let us do some word problems on multiplication). It is a little exposure and then practice of the narrowly understood procedure. It is not mastery with rigor.

4. Mastering a concept means, the student has the language, the conceptual schema(s) (effective and efficient strategies), accuracy and fluency in skills and procedure, and can apply it to other mathematics concepts and problem solving. The procedure of mastering multiplication tables should be based on solid understanding of the language and the concept. Students and the teacher should arrive at strategies and procedures by exploring and using the language, the conceptual schemas, and efficient and effective models. And then from several of these procedures should arrive at those that are efficient and generalizable (the standard algorithms). Students should develop, with the teacher, the criteria for efficient and effective conceptual schemas for deriving facts and procedures for multi-digit multiplication. The teacher should also help develop an efficient script for students to follow the steps needed to executeprocedure. Once children have arrived at an efficient procedure or procedures, they should practice it to achieve fluency and automatization. The fluency should be achieved by applying it in diverse situations. It means, ultimately, they have understanding, fluency, and applicability. Children learn tables successfully when teachers give them efficient strategies, enough practice in doing so and make it important to do so. They understand and are able to apply them according to how well they are taught.

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. Counting by 1, 2, 10, and 5. The instructional practice of having students count groups—skip counting—is an essential transition between additive and multiplicative reasoning. This counting should be limited to counting by 1, 2, 10, 5, and possibly 9. All other groups, when being added should be done by decomposition (adding 6 to 36 should be accomplished by asking: What is the next 10s? “40” How do I get there? “add 4” Where did the 4 come from? “from 6” What is left in 6? “2” What is 40 + 2? “42” So, what is 36 + 6? “42” Encouraging to count after 36 to add 6 does not amke the child acquire a robust numbersense. Just like visual clustering or representation of number as a group is a generalization and abstraction of discrete counting, skip counting, emphasizes the structure and efficiency that grouping gives to counting and, therefore, to addition. For example, counting by fives (using the fingers on hands as a starting model, then moving to TenFrame, Visual cluster cards representing 5, and then the 5-rod (yellow) of the Cuisenaire rods is the right progression for learning to count by 5. or twos (using eyes, or stacks of cubes, Visual Cluster Card representing 2, then the 2-rod (red) of the Cuisenaire rods) is very productive. Similarly, counting tens rods (in base 10 blocks or the 10-rod in Cuisenaire rods, however, using the Cuisenaire rods is better) as: 10, 20, 30, 40, and so on, emphasizes the concept of repeated addition and grouping. However, if these counting sequences are learned by discrete counting (Unifix cubes, fingers, number line, etc.) or without models to support the grouping and repeated counting activity then the order and the outcome will be learned without the concept and significant meaning about multiplicaiton or division.

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: repeated addition, groups of, an array, and the area of a rectangle; commutative and associative properties of multiplication; and distributive property of multiplication over addition and subtraction: a(b + c) = ab + ac and a(b – c) = ab – ac have been mastered. Multiplication tables should be mastered only after the groups of and area of a rectangle is clearly understood. If we use Cuisenaire rods for modeling multiplication, particularly for showing it as area of a rectangle, then the repeated addition and groups are already embedded in it and children can see the commutative, associative, and distributive properties also. Using these propeties, the teacher should derive multiplication tables up to 10 (i.e., 10 × 10 = 100 facts), in the following order (I cannot oveemphasize this order).

(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) Table of 10 (17 new facts), (iv) Table of 5 (15 new facts), (v) Table of 2 (13 new facts).

(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 = ?. Teacher: Do you know the answer? Student: No! Teacher: Which is the bigger number? Student: 8. Teacher: Good! Do you know 8 × 5? Student: Yes! Teacher: Good! Break 6 into 5 and 1. What is 8 × 5? Student: 40! Teacher: What is 8 × 1? Student: 8! Teacher: What is 40 + 8? Student: 48! Teacher: Now, what is 8 × 6? Student: 40 + 8 = 48. 8 × 6 = 48. First, If necessary, students form this fact as area of a 8 by 6 rectangle concretely with the help of Cuisenaire rods. 6 brown rods forming a 8 × 6 (vertical side = 8 and horizontal side = 6) rectangle and then breaking it into two rectangles (8 × 5 and 8 × 1). Students, now, derive these, by seeing the graphic organizer: 8 × 6 = 8 (5 + 1) = 8(5) + 8(1) = 40 + 8 = 48. Then, they should repeat it by visualizing it. All of this work should be done orally creating the script as described above. There are several ways the result can be derived by decomposition/recomposition.

5. Improving Times Table Fluency: The Institute for Effective Education (IEE) in the UK has published a new report on improving times table fluency, as a result of study of 876 children in 34 Year 4 (grade 3 in the U.S.) classes. All groups had similar pre-test scores and similar groups of children–same distribution of children with similar abilities. Each class used a different balance of conceptual nad procedural activities during times tables lessons. Conceptual activiities were games that focused on the conncetions and patterns in table facts, while procedural activiities were games in which students practiced multilication facts. All grous had same pre- and post tests. The report concluded that times tables may be best taught by using a balanced approach–teaching both the concepts behind them and practicing them in a range of ways with low-stakes testing.

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).

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 visual clustering(generalization of subitizing), decomposition/recomposition, comparison of numbers, and their relationships.

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: ‘The Number War 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^{[2]}.

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 number: e.g., five of clubs represents the number ^{+}5. Whereas, 5 of hearts represents, the number, ^{–}5. Thus, in both decks, when working with integers, red cards represent negative numbers, and black cards represent positive numbers. As one can see, numeral is a representation of quantity and number is a directed numeral (it has a direction and quantity). Up to fifth grade, we do not make a differentiation between numeral and number. However, once the children beginn to learn about integers, we need to differentiate between numeral and numeber.

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. There are a total of 45 sight facts^{[3]}. Sight facts are like sight words. A child should master these 45 sight facts by the end of Kindergatrten.

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 auditory form (f-i-v-e), and, (c) the quantity represented by the cluster. This integration is called “numberness.” In this particular case, this integration is called “fiveness.” Writing should begin when a child can recognize the cluster representing a numeral instantly.

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.

^{[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.