How to rectify the issues releated to writing in mathematics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Few mathematics teachers insist students to write on graph paper.

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

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

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

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

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

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

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

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

And, many more ….

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

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

4. Writing Process

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

• Sensory Motor and Visual Perceptual and Spatial systems

• Socio-Linguistic Systems

• Cognitive Processes and Executive Systems

• Social-Emotional Systems

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

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

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

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

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

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

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

(a) Fine Motor Skills and Writing

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

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

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

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

(b) Cognitive Skills and Writing

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

(c) Role of Equipment in Writing

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

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

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

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

(d) Role of Cursive Writing

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

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

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

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