Dysgraphia and Mathematics

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, lack of 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.

  1. 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.
  2. 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.
  3. 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.
  4. Aphasia: Omits and substitutes letters, symbols, numbers, operational symbols in writing, copying and communicating mathematics information.
  5. 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:  

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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 motor coordination 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 contrastingcomparing 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, then I 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, then I 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 ahalf circle/moon facing left’ by ending at the bottom of the place where we started and then go 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”:

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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, 23, 23, 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: 

  1. 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), 
  2. Opposite perspective (knowing the opposite perspective (to your right, to your left) (acquired around age 7-8), and, 
  3. 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. 

Improving Cognitive 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. 

  1. 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.  
  2. 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.
  3. 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.),  
  4. 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).’’ 
  5. 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!”
  6. 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?” 
  7. 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.
  8. 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: 

  1. Pencil grips or different types of pens or pencils to see what works best for the student.
  2. Typed copies of classroom notes or lesson outlines to help the student take notes in the package. 
  3. 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.
  4. 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.
  5. 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.
  6. Strategies and materials to address needs of left-hand writers. 
  7. Paper assignments with name, date, title, etc., already filled in.
  8. Proper, accurate, and clear instructions for tasks.  Make sure, students have understood the instruction before they attempt the problems. 
  9. Information needed to start writing and project assignments early. Help the students break project assignments into steps.
  10. Rubrics and explanations to show how each step is graded.

And, in the case of unique cases,

  1. 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. 
  2. Give examples of finished project assignments as models and demonstrations.
  3. 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:

  1. 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.
  2. 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. 
  3. 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.
  4. 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,

  1. Adapt a portion of test formats to cut down on handwriting. For example, use “circle the answer” or “fill in the blank” questions.
  2. Use a scribe or speech-to-text so the student can dictate test answers and writing assignments.
  3. Let the student choose either print or use cursive for handwritten responses. But, they must practice both.
  4. Allow a friend/classmate to “proofread” to look for ‘handwriting’ errors, not the conceptual and procedure errors. 
  5. 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. 
  6. 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.

Chung,P. & Patel, D.R. (2015). Dysgraphia. International Journal of Child Adolescent Health; 8(1): 27-36. https://www.eoepmolina.es/wp-content/gallery/Dyslexia-dysgraphia/Dysgraphia.pdf

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. ISBN 9780060162535.

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. 

Dysgraphia and Mathematics

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