Mastering the Concept of Number

Mastering the Concept of Number ​

Mastering the Concept of Number

Mahesh C. Sharma

Mathematics is a language and science of patterns. With this language
science, technology, and engineering are understood, expressed, and made

Development of numberness (quantity—number symbol relationships) is an essential component to successful numbersense and numeracy, and as student progresses to complex topics in mathematics. The basic math skills children bring to first grade affect their mathematics learning throughout elementary school years. For example, children who understand number, can go back and forth, easily and quickly, breaking (decomposition/recomposition) a larger numeral into smaller parts and putting back together have a head start in numbersense. First graders who understand the number line and can place numbers (at least up to 30) on the line (when dictated a series of numbers randomly) and who know some basic facts (sight facts) show faster growth in
math skills than their counterparts. These are the factors that make a difference in the first grade beyond intelligence and other abilities.

Particularly, when intelligence, working memory and other abilities are factored out to determine the most critical beginning of school math skills. Number concept is the organizing principle for developing numbersense. However, as mentioned earlier, number concept (numberness) is more than rote counting and even comparing the sizes of numbers representing
collections. It is important, therefore, for parents, curricular planners, and teachers to know the nature of the beginning of school knowledge needed to learn mathematics. In order to improve basic instruction, we have to know what to instruct and how to instruct. Mathematics knowledge is incremental, and without a good foundation, a student will not do well because
mathematics gets more complex.

What does it mean to know number concept

I. Reading vs Arithmetic: Phonemic Awareness and Numberness
To become fluent in reading, one needs, among others, two important processes: Phoneme awareness and completion tasks. Phoneme awareness (PA) is the ability to segment a spoken word into its separate speech sounds. It is the chunking and blending required for decoding or uttering a string of memorized letter-sound links. Completion tasks involve trying to recognize the word from the partial information uttered. In this, the brain jumps to a pronunciation when it completes the partial information into a recognized word. Both are the outcome of well orchestrated, supervised practice. Thus,
phonemic awareness, a good repository of sight words, ‘chunking and blending’ with the help of PA and sight words, and consistent supervised practice set the child on the way to becoming a reader.

Our increasing knowledge of phonemic awareness (PA) and its importance to the development of strong reading skills may inform our understanding of numberness as a basis for developing strategies to help children acquire numbersense. Just as phonemic awareness is critical to learning to read, i.e., to the teaching of reading, the building of numberness/number concept is critical to number relationships and arithmetic instruction and it benefits all students with or without learning disabilities. 

The question is: 

How can we best help children master this difficult hurdle?

Just as sounding out each letter cannot make the child a reader, the common practice of recognizing a collection of objects by one-to-one counting, arriving at arithmetic facts by sequential counting on number line, hash marks, or counters, doing number drills, and flashcards (and their modern versions on the computer or the use of iPad apps) cannot help all children arrive at numberness and arithmetic facts. It has not worked for most children. Research has shown it, teachers have seen it. We need to look at it from fresh perspective. It is a national and international problem. We need to learn from what has research shown in teaching reading during the last thirty to forty years. The understanding of learning problems in mathematics is in its infancy. To answer the question above, we need to focus on these questions:

(a) What is the nature of number conceptualization? 
(b) What is the trajectory of the development of number concept? 
(b) What can we learn from the research in science of reading? 
(c) What are the implications of this information to teaching number concept? 

A. Parallels: Letter Recognition and Number Concept

Before identifying the behavioral and cognitive markers for number concept, we need to examine the behavioral and cognitive markers for recognizing a letter

1. Knowing Letters and Reading Skills

We say that a child has the mastery of a letter when he shows the following
behavioral/cognitive markers:

(a) Can recognize the letter instantly (e.g., says: “EM” when he can recognize
the standard form of the letter M, instantly),

(b) Can recognize the letter in its varied forms— different and variant, or even
slightly deformed (e.g., M, M, M, N, M, M, M, M, M, etc.),

(c) Can recognize the letter in the midst of other letters and symbols (e.g., M
in the midst of other letters in CALM, MILK, WARMER, $M$569A, etc.),
(d) Associates a sound to the letter (e.g., the sound of the letter is like M in
Monkey), and

(e) Can write the letter without any prompts (e.g., can write M on an
imaginary board with eyes closed and can describe the writing process—
delineates the various strokes of the letter in the proper order).

When the child demonstrates these markers for all the letters of the alphabet, with the same level of mastery (it does not matter whether the letter is A or Z, lower or upper case), then we say that he has mastered the alphabet information. Once, the child knows then reading skill is acquired by the integration of the following

(a) Letter Concept,
(b) A large sight vocabulary,
(c) Phonemic Awareness (phoneme/grapheme connection),
(d) Chunking and blending using (a) —(c) (“chunking” of a new and big word
into smaller sound components using acceptable rules, and “blending” of
smaller word/sound segments into bigger words using effective strategies),
(e) Supervised and individual practice.
Integration of these requires deliberate practice. This is achieved by multiple, repeated, exposures to quality instruction based on the science of reading,

organized by knowledgeable and sympathetic adults and practiced by the child, both supervised and individually.

There is similar process in developing the idea of numberness (number concept) and then numbersense. A child has to show similar mastery markers (behaviors) for the first ten counting (natural) numbers and zero. Just as the understanding of phonemic awareness has revolutionized the teaching of beginning reading to children, the influence of proper development of number concept (numberness) carries implications for instruction not only for children who struggle with mathematics but for all children. The problem, however, is that mathematics educators and psychologists have not clearly defined and understood the meaning of number concept or numberness, in particular, and numbersense, in general. Because of this limited understanding, they have failed to clearly define, assess its absence, and, therefore, effectively teach it. The methods of teaching number are thus limited to the definition of “knowing” number (for most people, it means one-to-one counting and recognizing and writing numerals). Therefore, rather than focusing primarily on what students need to know, we should also focus on how they effectively learn number concept.

Understanding exactly how numberness and automaticity of number facts develops — thereby opening the door for higher order fluency and comprehension, is the key. That will provide for an effective numeracy pedagogy, not for those having difficulty, but for all children. Fortunately, research in cognitive science has already identified powerful principles of
learning to help us get to some answers for these and other important questions in mathematics instruction.

The objective, therefore, is to develop and focus on the clear understanding of numberness, number concept and numbersense. Let us begin with the end in mind. Children with good numbersense can

• move seamlessly between the real world of quantities and the mathematical world of numbers and numerical expressions;
• represent and use the same number in multiple ways/forms depending on the context and purpose;
• combine numbers with ease using effective, efficient, and elegant strategies;
• recognize benchmark numbers, compare numbers, and number patterns;
• recognize gross numerical errors and have a good sense of numerical magnitude; and

• think or talk in a sensible way about the general properties of a numerical problem or expression, without doing any precise computation. 

In order to operationalize the definition of numbersense for instruction, we should keep in mind the parallels and differences in the nature of reading skills and numeracy, particularly, phonemic awareness and number concept/ numberness.

2. Knowing the First Ten Counting Numbers:

(a) lexical entries for number (having a good store of number names —can recite number names in order, knows the difference between number words and non-number words, e.g., “seventy-three” is a number word, but “seventy-cuke” is not a number word), just like children can fluently recite the letters of the alphabet in order;

(b) can meaningfully count (one-to-one correspondence + sequencing = conservation of number)—can assign a number to a collection of objects


(c) instantaneously recognize visual clusters (extension of subitizing) of numbers up to ten,

(d) can represent quantity of the collection in its orthographic form (giving a number name to the visual cluster and write the number —a visual/ graphical representation, e.g., 5), and can assign a name to the graphical representation and the sound (say the number 5, e.g., f-i-v-e), and

(e) decompose and recompose a collection (number)—knowing that 5 can be expressed in all possible combinations, e.g., 5 is 4 plus 1, 1 plus 4, 3 and 2, and 2 and 3.

(f) can form number shapes properly. Just like writing letters, where we emphasize proper formation, we should emphasize the proper formation of numbers.

It requires deliberate practice in the integration of these elements to achieve numberness, particularly, the decomposition/recomposition process. Similarly, a collection of seven objects should visually invoke in a child’s mind: (a) the collection representing that quantity, (b) the number name, (c) its visual and auditory representation, and (d) the different combinations of numbers, e.g., 6 +1, 5 + 2, 4 + 3, 3 + 4, 2 + 5, and 1 + 6. On the other hand, the number ‘7’ should invoke the idea that it represents seven objects and a word that represents it. Similarly, on hearing the number word ‘seven,’ one
should instantly associate a collection of seven ‘non-specific’ objects and the visual representation ‘7’.

This integration of visual clustering (cluster of 7 objects), graphical representation of number (7), decomposition/recomposition of number 7 (breaking 7 into its sight facts: 1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2, and 6 + 1), and the associated phoneme to that number (saying the number s-e-v-e-n) is called numberness.

Presence and integration of the skills from (a) to (e) above for number 7 is knowing the number 7. These are the behavioral and skill markers for knowing number 7. Fluent integration of these markers is true numberness, mastery of number 7. This is essential to understanding what the number 7 means. This should be true for all the numerals from 1 to 10.

In reading, phonological decoding associates graphemes and phonemes, but in arithmetic, numberness is the integration of three elements rather than two. It is a process as shown below:

As the diagram illustrates, the process is not a one-to-one linear interaction but consists of simultaneous multi-interactions. The three processes

recognition of numberness (visual cluster) and graphical and phonemic representations of number should happen instantly.

3. Implications of The Parallel to Instruction
The development of numberness (particularly the decomposition/ recomposition of number), numbersense and proficiency in arithmetic (numeracy), on one hand, and phonemic awareness and the ability to read with understanding and purpose, on the other, are comparable activities.

Numberness (particularly, the decomposition/recomposition of number) is to arithmetic as phonemic awareness is to reading. Nevertheless, it is also important to recognize the differences between the two processes. In reading, phonological decoding associates graphemes and phonemes, but numberness is the integration of three elements: grapheme (visual representation of
number), phoneme (saying the number), and the numerosity (visual cluster representing the quantity). Associating, quantity to grapheme and phoneme is the difficult part

The experiences, skills and knowledge that children bring to school place them at different starting points, suggesting they might need differing amounts and/or pace of instruction. But, that instruction should be quality instruction, with the same goal for all children, with or without learning disabilities. In our special education programs, generally, the strategies used to teach
number are inefficient. They are mostly counting based. To drive even the sight facts, it is not enough merely to know the counting rules. Research and experience tells us that children with numbersense not only have efficient counting skills but go beyond counting to recognize numbers automatically. Visual clustering and decomposition/recomposition shifts the cognitive load from focusing on each number as the result of a count to number relationships quickly in order to acquire sight facts and then arithmetic facts. Children who develop automatic integration of these skills can move on to focus on fluency and comprehension while our special education and many other children continue to struggle, laboring at the counting level.

B. Number conceptualization: Numberness/numerosity
To most people, knowledge and use of the first ten natural numbers (1, 2, 3,…, 9, and, 10) appear to be a simple and straightforward process. To them, learning to count is merely a matter of reciting a string of words like a nursery rhyme, a feat that most young children can master surprisingly early and easily. Most children are able to progress naturally from working with objects and then to represent these experiences in pictures and icons, and finally they represent and manipulate numerical symbols for objects and collection of objects.   

Early humans observed that three apples and two apples make five apples, and also that three and two sheep make five sheep; and then they were able to make the important leap: that the ideas of threeness and twoness, whenever combined, result in fiveness.  The numbers 3, 2 and 5 themselves do not physically exist in the real world, but they describe
how many objects we may have in some set. The important thing is that we can also reason about them, independently
of apples or sheep. This generalization—going from concrete representation of objects to an abstraction, is the basis of number concept
Cognitively, it is a higher skill and concept than just counting objects.    As letters are represented orthographically (graphic), verbally (the name of the letter), and the associated contextual sound, similarly, numbers (orthographic images) were invented to represent the numberness/numerosity and to communicate information about quantity. 

Numbers can be represented in three main formats: Hindu-Arabic orthographic form (grapheme—numerical format), verbal form (phonological word), and magnitude-related (Dahaene, 1992; Fries, 1996). Learning number means functional relationships between these different representations and their processing characteristics. Of particular interest is understanding magnitude information because magnitude information is the 

semantic aspect of numerical processing. This is so because each number, whatever its format, is a symbolic representation of a magnitude, quantity, or measurement. This understanding sets apart literacy and numeracy skills. The quantities represented get more and more complex and the numbers that represent them also become more and more complex as we move on the journey of learning mathematics.

The magnitude-related representation is the most difficult one and the absence of this creates number related difficulties, such as dyscalculia. To facilitate this, we want to introduce the initial representation of quantity in the form of visual clusters. Visual cluster representation subsumes and extends the subitizing (knowing a collection up to 4 – 6 by observation), on the one hand, and the magnitude of number, on the other. Thus, understanding of the representation of number is the integration of (a) Hindu-Arabic representation (orthographic/graphemic), (b) verbal (phonemic) and (c) visual cluster representation of the number (knowing the magnitude of the number).  An average child in a normal course of the learning process takes around five years, from about age two to six, to learn to handle numbers and to apply them to everyday situations to solve simple quantitative problems accurately and consistently. Yet many children have difficulty in mastering and applying this skill according to a socially acceptable timetable or an acceptable level of mastery. For a variety of reasons, this process may be longer and difficult for some of them. Because of inefficient methods of teaching number, both formally and informally, many children do not become secure in their ability to read and write numerals, to visualize sets of objects or in their sense of number and their applications.      


A.  Lexical Entries: Number Names and Counting

Most children come to school reciting the string of the alphabet as a song—a rhyme. Similarly, long before school, many children begin counting by rote— the recitation of an ordered sequence of responses using concrete objects. In many cases, this counting does not mean real understanding of number or its effective use. It is acquired by observing others using a series of number words. Exposure to other people counting or another child’s counting adds to the development of a larger vocabulary for number names. Just like a child should know the letters of the alphabet by rote first and then the meaning, similarly, children should acquire the string of number names, particularly to about 30 to 40, preferably 50, before they come to Kindergarten. 

1.  Counting 

Counting by rote is an indication of the presence of lexical entries relating to number (number words). Though these entries are essential, they are not sufficient for fluent number conceptualization and number usage. Counting as a rote activity can be taught easily, but the child may or may not know what he is doing except saying the words in a sequence—a string of words. Let us consider the following example of a four-and-half year child working with number. 

Teacher: Can you tell me how many cubes there are on the table? (Points to a collection of cubes on the table.)


Child: (Child sequentially counts by touching each cube once) Seven.

Teacher: When you were counting, what number came just before seven?

Child: I do not know. Is it one? Three? Five? I don’t know. 

The child thinks and says: “Lemme see.” 

Teacher: Please, count the cubes again. (Points to the same collection.)Child: (The child counts the cubes again using the same strategy of touching each one as he counts them.) Seven. Teacher: Can you give me six cubes?

Child: I don’t know. I don’t think I have enough here. Do I have enough to give you? Maybe I do. (He counts six cubes one by one and gives them to the teacher.)

Teacher: That is right. (She gives them back to the child.) Please put them back the way they were. 

The child rearranges them. 

Teacher: How many cubes are there on the table now?  

The child counts them by touching each one once and answers.

Child: Seven. 

Pauses and says: “Oh! Seven again. The same seven, I guess.” Teacher: Yes!

This example shows that the child knows the number words but does not know what those number words mean. He has learnt these number words by rote. Thus, rote counting may not help a student in problem solving such as the comparison of two sets or operations on numbers. Though this rote counting is not true numbersense, it is a starting point for number conceptualization. This rote counting process is essential as it develops the lexical entries for numbers. However, with rote counting as the only skill, children may not use the simple one-to-one counting as a strategy to compare two sets even when they are given perceptual reminders of this correspondence. To extend the lexical entries for number, it is important that teachers devote a few minutes each day to sequential counting.  This helps children to see the structure and patterns of numbers. 


2.  Egocentric Counting to the Cardinality of the Set: Numberness To be effective and meaningful, counting behavior must have the underlying cognitive structures and processes and the support of language. Many children of ages 4 through 6 count objects in a rote manner, and for them, the last number uttered represents the “cardinality of the set.” For them, the cardinality of the set is the outcome of their counting process, not the property of the collection. They think of this number as the outcome of their action (“These are six blocks because I just counted them.”) rather than the affirmation of the fact that the quantity representing that collection—a property of the collection (“These are six cubes.”). This is a key transition in number conceptualization. The following illustrates how to achieve this transition. Let us consider the example of an interaction with another child during number work in a Kindergarten class. 

Teacher: Can you show me your right hand? 

Student: With a little hesitation raises his right hand.  

Teacher:  Good! 

Teacher: Can you tell me how many cubes are there on the table? (Points to a collection on the table.)


Child: (Sequentially counts by touching each cube once) Seven.

Teacher: Very good!  How many did you say?

Child:  I think seven. Let me see.  (He counts again in the same manner.) Seven.  There are seven. There are seven, I just counted. Right?

Teacher: Yes! You are right. There are seven cubes. When you were counting, I noticed that you started counting them from left to right (points to the direction) what if you counted them from right to left? (Points to the direction.)

Child: I do not know. Let me try.   

The child thinks and counts.

Child: It is seven.  

Teacher: That is right.

Child: (As the teacher acknowledges child’s answer, the child is thinking about something and then touches the cubes. The child counts the cubes again using the same strategy of touching each one as he counts them, first from left to right and then right to left.) “Seven. It is the same thing.  It’s seven. It doesn’t matter how you count; it is the same number. It’s seven. See!” (Counts again once from left to right and then right to left.) “I guess it is always seven.  There are seven cubes.” 

In the first example, the child is only reciting a string of words.  He is not connecting the different number words with each other and with the property of the collection. The number (cardinality of the set) arrived at by counting only shows that the child thinks the number is the property of the counting process not of the collection. This is called egocentric counting, when the child thinks that the  number produced is the outcome of that counting. Many young children when asked to count the same collection again may produce a different answer and may not even question themselves. Most children with exposure to counting, problem solving, and discussion transcend this type of egocentric outcome of the counting process. 

In the second case, the child acquired an important concept: the number (cardinality of the set) is not the property/function of the counting process but the property of the collection. 

Associating a number to a collection and considering that number as the property of the collection is the first step in the direction of conceptualizing number properly. The type of questions asked in these two activities are the means to converting a child’s concrete experiences and egocentric counting into abstract concept—numberness (number represents the cardinality of the set—it is the property of the set not of the counting process).  Above examples illustrate how the structure of a problem and nature of questions asked influence the strategies and concepts children develop to solve problems.


B. Concept of  Subitizing and Visual Clusters

The three representational codes (verbal, Arabic, and magnitude information) each constitute the starting point for different arithmetic activities (Dehaene, 1992).  Concept of visual clusters is just like recognizing letters and words, instantly.  We have a mental orthographic/graphemic representation of a word

(It is not in a particular font or size—it is a fluid and malleable representation). It is stored in our mind. When we see a printed word, we 


match it with the stored abstract orthographic representation. We “see” the “image” of the word in our mind’s eye when we
activate this representation at a meta level.        

Just as orthographic patterns of letters are abstractly stored as “mental orthographic image” (MOI) or “mental image of the word,” we form the image of a collection of objects in our brain and then try to associate mental orthographic abstract image of the number associated in our brain. Most children can experience the ‘image’ of the letters of a printed word in their mind’s eye, however, it needs practice. If one can form mental orthographic image of letters and words, then one can also understand and embrace the concept of visual cluster. With practice, we should be able to see the image of the cluster of objects in the mind’s eye. Educators need to make a more direct connection between the “image” a child experiences when she activates the underlying mental graphemic representation at a meta-level and what they do in their instructional/clinical practice of teaching number. Children taught with an emphasis on counting do not form the image of clusters and therefore, have difficulty automatizing sight facts and later other arithmetic facts.

The ability to visualize in general (for example, I can visualize my kitchen in my mind’s eye as I sit here in my office) is
separate from the ability to develop, store, and retrieve mental graphemic representations (MGR). It is the familiarity and practice that helps the child in forming the image. In fact, we should begin visual cluster activities using that experience of visualizing a familiar scene (their kitchen, classroom…) to help them understand the concept of seeing something in their mind’s eye.  The brain, as it learns to speak, begins to regard words as single sounds (processing the whole rather than the parts—it does not seem to them as made up of parts). However, when a child learns to read, the brain has to reprogram these pathways in order to be able to identify the individual bits in spoken words that are represented by letters/phonemes. Brain can perceive and recognize speech sounds does not mean that four or five year old children can naturally identify the

individual sounds in words. Therefore, we have to teach them to identify and manipulate these sounds.  This process is
new to them.  It is different from infants learning to speak. The brain has to learn how to identify these bits in order to link them to a visual shape—a letter. This is essential new learning that is part of the new skill we call writing and reading.  

The process is not very different in learning number.  Initially, the child sees the five fingers of one hand as separate entities and adults help him to count them to find out how many. They provide the names as he counts. Eventually, the
child sees the five fingers of the hand as a collection, not made up of components.  We teach children it is made up of five individual fingers. But, with experience he knows it instantly that each hand has fie fingers. That is fiveness. Up to this point, there are similarities between reading and arithmetic.  The extra complicating item in arithmetic is that it is cumulative in the sense that every number (as a representation of quantity) also subsumes previous numbers (as a combination of collection of quantities). A collection of five also has a sub-collection of 4, 3 and 2, and 1. The development of these skills calls for effective methods of teaching.  In that sense, is more like a word than a letter: not only the collection of five individual objects is 5, but it is also made of smaller numbers. This is another important abstraction that is new to him. 

So the question is how do we teach this collection of new skills?  If it does not come naturally to all those struggling children, what are the best ways to call their attention to the fact that the words and numbers they say are actually not a single bit, but strings of different bits?  The brain has to be trained, it is dealing with this new set of skills.  

The alphabet was invented to make speech visible. Similarly, number names and their representations were developed to make the quantity (magnitude) visible when the collection is not there. In teaching number, we duplicate one of the most
fundamental flaws found in almost all phonics instruction today, including  traditional ones, is that they teach the code
backwards. That is, they go from letter to sound rather than from sound to letter.  The print-to sound (conventional
phonics) approach leaves gaps, invites confusion, and creates inefficiencies.
We create the same problem in teaching number—most teachers begin teaching
number from number shapes to quantity. 
It should be from quantity to orthographic representation. 

As in reading, there are two elements involved—the speech-bits and the sound-bits. In learning number there are three elements involved: recognizing a cluster of objects as a whole, the orthographic image, and the spoken name for the number. We need to help them become conscious of the two elements in reading and the three elements in learning number. However, you cannot accurately visualize the printed word in your mind’s eye until you have developed a fully specified, “robust” storage of the mental orthographic image  

word form (MOI) for that word. These fully specified MOIs develop gradually (some more gradually than others) with repeated, meaningful exposure to the printed word and the related analysis—chunking and blending, seeing similarities and differences, visual image of the word, etc. So at any given point in time even a typically developing reader/speller will not be able to visualize a particular word without enough practice.

1.  Visual Clustering

There is abundant research which tells us that struggling readers and writers (spellers) may not develop fully specified MGRs as easily as typically developing students and so we cannot a priory assume that most children can visualize what the (printed) word looks like over the population of struggling readers and writers (spellers).

Some students need to be reminded to take the time to think about what the word looks like. This is where teaching students the meta-linguistic strategy of how to use their MOIs – once fully specified – is important. So, when we think about intervention for students who have “fuzzy” MOIs, it is twopronged: activities that help students establish robust MOIs and activities that teach students how to apply these MOIs when creating a visual cluster in the mind.  Its application to decomposition/recomposition to learn sight facts will come automatically because once a student has developed a fully specified robust MGR, she will automatically recognize a cluster when deriving arithmetic facts.

Counting objects is a natural activity for children. They are exposed to counting in abundance—at home, in school, in their games and playing with toys.  However, for proper number concept, the child must master and transcend counting. Just as many teachers and the curricula used to focus on letters only rather than phonemes and graphemes, similarly, many teachers and curricula keep on focusing on counting. Letters are not the basic units for decoding, sounds (phonemes) and their spellings (graphemes), similarly, instead of focusing on “counting up” for adding and “counting down” for subtracting, instruction must focus on strategies derived decomposition/ recomposition of visual clusters for “sight facts” for for numbers up to 10. However, understanding the processing details of visual clusters (magnitude or quantity) in relation to Arabic and verbal representations is an imperative. Forming an image of the magnitude plays an important role in developing numberness—estimation, counting, number comparison, and mental arithmetic. 

Access to magnitude information represented by a number is necessary before any other numerical processing can take place (McCloskey, Caramazza, & Basile, 1985; Mccluskey, 1992). But, it should transcend 1-1 counting.  It should be in terns of patterns, like visual clusters (dominos, dice, die, playing cards, VCCs, etc.). Introducing children to visual clusters facilitates the other representations and children take off using any of the three representations.  And during processing, information can be exchanged between the systems.  In other words, they can go from visual clusters to cardinality of the set to location of number on number line. 

The ability to connect language to images in the mind (visual/verbal integration) seems so basic that it is often assumed that everyone does it. However, significant improvement in language comprehension occurs when mental imagery is stimulated, suggesting that forming and using imagery is not automatic for many people. Similarly, carefully sequenced activities about number visualizing and verbalizing using Visual Cluster cards (VCCs)) can help children hold clusters in the mind’s eye, therefore, numerosity of the collection in the mind’s EYE. This provides a helpful framework for stimulating the ability to “see with the mind’s eye” resulting in forming number relationships and transcending counting. 

After initial exposure to counting materials of different types (e.g., counting blocks, hash marks, number line, fingers, pennies, buttons, marbles, etc.), materials such as dominos, dice, and playing cards are better models for the development of the number concept. Visual Cluster cards are the most effective instructional material that helps develop the number concept– sequencing, one-to-one correspondence,  visual clustering, and decomposition/recomposition.[1]

Children with or without mathematics difficulties or risk for number difficulties and even those who experience difficulty with written and/or oral language processing can learn to use mental images of visual clusters to get to what they already know about certain numbers and apply that information to what they need to understand in number relations. The human brain’s nature and ability to see and visualize patterns is the principle behind these cards. As persons gain skill in the process of visualizing and then verbalizing about what they see, they begin to independently match conscious images of clusters (gestalt of the cluster) to the concepts (numbers) expressed by a group of pips forming the clusters. An example of a “pip” or “icon” on a card.  The card representing number 1 will have one pip on it, in the middle of the card. 

A deck of Visual Cluster Cards (VCC) consist of 4 cards with each with 1 pip to 10 pips in 4 suites (heart, diamond, club, and spade); a card without any pips–represents the number zero; 2 jokers that can be given any value during card games such as “Number War Game.”[1] Cards showing 3, 8, 9, and 10 have two to four types of representations of pips on them (in the standard form

[1] For number games see: Games and Their Uses for Mathematics Learning  by Sharma (2008) (CT/LM)

of clusters with slight variations and alternative arrangements). For example, below, the clusters of three pips representing number 3, in two arrangements. 

2.  Decomposition/Recomposition

The pips on number cards (e.g., 2 through 10) are organized in such a manner that the sub-clusters of smaller numbers than a given number can be instantaneously recognized on the same card.  For example, on the 7 of diamonds in the VCC collection (shown below), one can see a cluster of ‘5 and 2’ or ‘6 and 1’ and ‘4 and 3.’ The diagram, below, demonstrates the relationship between the visual cluster of number 7 and the sub-clusters representing smaller numbers that make 7 (as 3  + 4, 4 + 3; 5+ 2, 2 + 5; 6 + 1, 1 + 6). Recognizing visual clusters on these cards develops in children the key concept of numbersense: decomposition/recomposition skill and the sight 

facts.  Decomposition/recomposition is responsible for developing the flexibility of use of numbers in arithmetic operations. 

The skilled questioning of a professional can cause the individual to notice where his or her images of clusters do not match the displayed clusters so that needed adjustments in images (and understanding) can be made. When images can be clearly organized in the mind’s eye and described verally, without looking at the cards, then decomposition/recomposition can take place and higher order tasks such as generalizing, making inferences, or drawing conclusions about number relationships become possible. Then, children are able to automatize addition and subtraction facts with ease. 

The patterns on the cards create a very rich, vivid, complete imprints of visual clusters. The power of imagery enables a person to see these visual clusters. What is taking place in this process is an example of eidetic memory.   By the help of these clusters, a child can recall the sub-clusters and form sight facts quickly and sometimes with only a little prompting. It is similar to learning sight words with visual exposures. This means that the student who could not master simple addition and subtraction sight facts without spending hours counting on fingers, number line, hash marks, and objects and re-counting can learn first the sight facts. Later, using decomposition/recomposition, making Ten, and mastery of teens numbers, they can expand sight facts to other facts.  Children who are taught number concept only through counting—counting discrete objects, number line, fingers, hash marks, etc., struggle with number fact recall, as they have not formed these rich, complete, eidetic imprints.  Most of the interventions, generally, focus on counting, speeding up recall (flash cards, MadMinutes, computer, iPAD, iPhone Apps) and some cognitive processing cannot achieve what children can achieve with Visual Cluster cards.  The efforts made and time spent on the use of multi-sensory aspects of VCC and Cuisenaire rods help create deeper, richer eidetic imprints.     

Research (Rayner, 2014) has shown that skilled readers of English language typically look at words for about 250 milliseconds, make regressions (e.g., look back at words) 10% to 15% of the time, and skip about 25% to 33% of words. Poor readers, in contrast, spend more time on individual letters in the words, regress more, and skip fewer words.  Similarly, children with little fluency in arithmetic facts focus on every number (as they count), whereas children with fluent arithmetic facts do not focus on each number but focus on clusters of numbers and use strategies derived from decomposition and recomposition of numbers. This suggest that it is particularly important for elementary- and middle-school teachers to select appropriate materials so children can progress steadily rather than risk slowing their fluency and understanding/comprehension development. 

All readers must learn the sound/symbol correlations that make up the written code in order to read and spell.  Intuitive readers mostly pick up these patterns through exposure and some direct instruction while those with dyslexia need overt, intensive, multi-sensory (see it, say it, write it) instruction in order to internalize the code.  Of course, meaning and context matter. Similarly, all children must learn the quantity/symbol correlations that make up the numeracy code in order to become fluent in mathematics.  Most above average and many average children pick up the patterns of number relationships through exposure and some direct instruction while a large number of average and below average children and those with mathematics difficulties such as dyscalculia need overt, intensive, efficient, multi-sensory (construct it, touch it, manipulate it, see it, visualize it, write it) instruction in order to internalize the quantitative relationships—number concept, arithmetic facts, place value, and arithmetic operations. These are better facilitated by effective and efficient materials and a well-thought out sequence of activities that are easily converted into “working scripts.” 

Multi-sensory teaching applied in the context of mathematics teaching and intervention means making mathematics concepts transparent, easy to understand, and helpful in arriving at the conceptual schema and procedure efficiently. It is a misunderstood concept by most educators. To many teachers concrete materials— “to touch, see, manipulate, and learn” objects. And, most of the time they end up counting objects and finding the answer with them. The purpose of concrete materials is arrive at efficient and elegant strategies. True multi-sensory apparatus implies that one is working on improving information processing based on sensory pathways rather than just improving the ability to associate quantity and symbols. 

Instructional approaches that are effective and efficient use direct, explicit teaching of quantity-symbol relationships, number patterns, decomposition/ recomposition strategies, develop conceptual schemas, and meaningful number relationships, and provide a great deal of successful practice of skills and procedures, with and without problem solving.

No one learns to read English or similar languages without phonics instruction. The only difference is whether we teach it to them, or they teach it to themselves.  We all do some of the latter, and most of us need some or more of the former, but no one learns to read without phonics instruction.  Similarly, no one learns arithmetic facts and number relationships well without decomposition/recomposition. The only difference is whether we teach it to them, or they teach it to themselves.  We all do some of the latter, and most of us need a lot of the former, but no one learns numeracy fluently without decomposition/recomposition. But, some do not acquire decomposition/ recomposition without intentional, direct instruction. Formal, intentional instruction in decomposition/recomposition facilitates and accelerates learning number concept, arithmetic facts, numbersense, and ultimately, numeracy—a major goal of mathematics instruction in elementary school.  When a student in higher grades has not mastered numeracy, we need to make sure that he has mastered decomposition/recomposition, first. 

3.  Sight Facts

The Visual Cluster cards are organized in such a manner that the sub-clusters of the given number are easy to recognize on the standard cluster of that number. For example, when we begin with the 8-card and add one pip to it we have the 9-card (see three 9-cards; first 9-card is made by adding the new pip in the middle of the 8-card; if we add that pip on the top on the 8-card, we get the second 9-card ; and if we add the pip in the bottom, we have the third 9card in the sequence), now we have decomposed 9 into two numbers: 8 and 1 and one can easily see the sight facts: 8 + 1 = 9, 1 + 8 = 9, 9 = 8 + 1 or 1 + 8.  Similarly, if we see the sub-cluster of two pips on the bottom of the first two 9-cards in the sequence,   and the cluster of 7 above the sub-cluster of 2, one can see that the 9 is decomposed into two numbers 2 and 7 it gives us the sight facts: 7 + 2 = 9, 2 + 7 = 9, 9 = 2 + 7 or 7 + 2.   

If we look at the 9-card, where the one pip is added to the 8-card in the middle column at the top (see the second card in the sequence), we can see the cluster of the 5-card on the top and the 4-card below it on the 9-card and thus 9 is decomposed into 5 and 4 or 4 and 5 giving us the sight facts: 4 + 5 = 9, 5 + 4 = 9, 9 = 5 + 4 = 4 + 5.   

On the other hand, if we put the 5-card and the 4-card together, we recompose it into the 9-card.

The 9-cards, in the middle, can be seen as cluster of 3 on the top and then a cluster of 6 below the 3-cluster. To get the decomposition of 9 as 6 and 3 better, we form a new 9-card starting with the 6 card and adding a group of 3 pips to the 6-card. This gives a 9-card with a different arrangement of clusters. This 9-card (the last card in the sequence) has the 9 pips that are organized into 3 groups of 3s or can be seen as one group of 3 and a cluster of 6. Now 9 is decomposed into 6 and 3 and we get the sight facts: 9 = 6 + 3 = 3 + 6.   The different arrangements of the 9 into these three 9-cards facilitate all the possible decompositions of 9 and, therefore, all the possible sight facts related to 9 (9 = 8 + 1 = 1 + 8; 9 = 7 + 2 = 2 + 7; 9 = 6 + 3 = 3 + 6; 9 = 5 + 4 = 4 + 5). These sight facts relate 9 to all the smaller counting numbers. The same process can be applied for all the other numbers up to ten.  In order to derive all the possible decomposition and recomposition, the numbers 8 and 10 also have alternative cluster arrangements. 

Decomposition/recomposition and sight facts are at the heart of number conceptualization and learning arithmetic facts (particularly, mastery of addition and subtraction facts). The distributive, associative, and commutative properties of operations and their application are also dependent on understanding the process of decomposition and recomposition. 

The decomposition/recomposition of numbers is achieved by a careful introduction of cards, using Cuisenaire rods, and children practicing the process under supervision. Many above-average and average children achieve the decomposition and recomposition of number skills using many other materials, even counting materials, but many average and most below-average


and special needs children have great difficulty in arriving at the decomposition/recomposition of number using counting strategies and materials.  

1.  List of Sight Facts

There are more than two-hundred sight words (number depends upon the program being used) that children must acquire. Most schools and programs expect children to master between 35 to 75 sight words at the end of Kindergarten. Then with the help of sight words and phonemic awareness, they begin to “chunk” and “blend” and with practice (supervised and individual) they learn to read.  Sight facts play the same role in acquiring arithmetic facts. The following is the list of sight facts.  

The key element is the acquisition and application of decomposition/ recomposition process.  Without this process, children do not acquire fluency in addition and subtraction facts. Once children have the concept of number, the decomposition/recomposition process can also be accomplished and reinforced with Cuisenaire rods[1]. For example, the number 10 can be shown as the combination of two numbers as follows (the same process can be used for all other numbers 2, 3, 4, 5, 6, 7, 8, and 9): 

Once children have formed these combinations (all the possible sight facts of10), the teacher helps them to make these combinations fluent and provides opportunities for applying these sight addition facts. When children have learned the sight facts of a number—recognized the combinations of sub-clusters on Visual Cluster Cards, formed them using Cuisenaire rods or InVicta Balance, can recite them, then they should be asked to record them with the help of these materials. When  children can supply the decomposition/recomposition of a number in several forms, they are ready to write the sight fact equations as pointed above in the sight fact equations for number 10. 

Repeated exposures to making combinations of numbers (sight facts)  by using Visual Cluster Cards and Cuisenaire rods are important as a starting point for learning other arithmetic facts. Children should move from oral to written form with specific, positive, and corrective feedback both for making combinations and acquiring fluency. The practice should involve only strategies using decomposition/recomposition[2].

When students arrive at arithmetic facts and procedures with the help of strategies they develop mathematics conceptual understanding with robust structures behind them rather than learning isolated facts and routine procedures. Knowledge structures here refer to conceptual schemas that students use to organize and relate language, concepts, and facts. Experts have developed complex knowledge structures with multiple and flexible

[1] For derails of how to use Cuisenaire rods see Cuisenaire Rods and Mathematics Learning (Sharma, 2013).

[2] Strategies for Teaching Addition Facts in How to Teach Arithmetic Facts Effectively and Easily (Sharma, 2008) 

interconnections based on fundamental concepts while novices and poor students have inefficient, simpler, disjointed knowledge structures with fewer connections that make it difficult for them to assimilate new concepts or ability to recall these facts. For example, they may think addition is just “counting up” and subtraction is “counting down.” These children end up working harder with little or no pay off.   

When a child does not have efficient strategies, it is important that we help them develop these strategies.  This requires timely and effective interventions.  As soon as a teacher observes that a child is having struggle in numberness, she must arrange for interventions.  All children benefit from effective math interventions. The quality of instruction and intervention is dependent on the competence of individual teachers. Teacher certification for pre-K through 3rd-grade should, therefore, emphasize both mastery of the content knowledge of the subject (specifically, a deeper knowledge of the arithmetic—number concept, numbersense, and numeracy) and strength in the mathematics content specific pedagogy. 

What we now know is that mathematics instruction—initial and intervention —is far more effective when delivered by a teacher who understands both the subject matter and the most effective ways in which young children learn math. Because the conceptual complexity of elementary mathematics is underrated, a successful program will ensure that early math instructors specialize in these areas. One solution may be for a school to designate a teacher in each grade who is responsible for teaching only math to all students of that grade and should provide  quality interventions to children who need it at that grade level.

Early instruction with quality activities that develop a comprehensive number concept and numbersense can minimize and prevent failure in numeracy and in later mathematics. For example, teaching the integration of numbersense activities with an increased focus on “sight number facts” automaticity will better prepare children for numeracy activities.  Teaching these skills in isolation and without effective strategies has minimal effect in the reduction of difficulties in mathematics for the general and LD population. Quality instruction benefits students with and without learning disabilities. 

Early learning mathematics experiences, can definitely help teachers implementing the position of the National Council of Teachers of Mathematics (2016), National research Council recommendations, and Common Core State Standards (CCSS-M, 2010).

In my extensive work with children with and without learning disabilities, the Visual Cluster Cards and Cuisenaire rods are the most effective materials to achieve decomposition/recomposition skills and mastering arithmetic facts.

One begins with VCC and then moves to Cuisenaire rods. The next chapter describes the development of the visual clustering and decomposition/ recomposition process using Visual Cluster cards and Cuisenaire rods.  With true numberness with decomposition and recomposition as the basis, one can easily learn the arithmetic facts, particularly, addition/subtraction facts. The concept of visual cluster and the decomposition/recomposition is the phonological equivalent of numbersense. Weaker a child’s decomposition/ recomposition skill and automatic fact recognition skills, the more the child has to rely on inefficient strategies such as counting.  And the compensatory use of inefficient strategies will never entirely make up for weak decoding, and even the most adapt compensator remains at an enormous disadvantage. Similarly, rote practice of arithmetic facts and procedures is not productive.  Practicing a skill over and over in the same way may teach students to acquire the skill, but it won’t necessarily lead them to apply that skill to other contexts. Rather, students need to practice the skill in a variety of different ways using efficient strategies to be able to retain, generalize and apply that information.

C.  Social and Academic Expectations about Numeracy

1. What is Numbersense?

To be fluent in numeracy, one of the components is having good numbersense. One’s open, positive, and flexible attitude toward number and the ability to display that flexibility and proficiency in handling quantity is called numbersense. As a skill set, numbersense refers to a person’s ability to look at the world quantitatively and make quantitative and spatial judgements and decisions using mental calculations using the properties of number. 

Technically, it is the integration of (a) number concept, (b)  number relationships (arithmetic facts), and (c) place value. It is age and grade specific. It gets more and more complex and sophisticated as a person encounters new number systems. For example, at the end of Kindergarten the expected mastery of numbersense is: (a) Number concept, (b) 45 sight facts, and (c) place value of 2-digits. Similarly, mastery of numbersense expected by the end of first grade is: (a) Number concept, (b) 100 Addition facts (sums up to 20), and, (c) Place value of 3-digits. And, numbersense, at the end of fourth grade is (a) number concept, (b) All arithmetic facts (addition, subtraction, multiplication, and division), and, (c) Place value up to the hundredth place. 

Numbersense consists of a cluster of ideas such as the meaning and ways of representing numbers, relationships among numbers, the relative magnitude of numbers, and proficiency in working with them (ultimately leading to mastering arithmetic facts and their usage). Number sense is not a set of discrete skills but a set of integrative skills. Students with good numbersense can move effortlessly between the real world of numbers and formal numerical expressions. They can represent the same number in multiple ways depending on the context and purpose. In operations with numbers, children with a good sense of number can decompose and recompose numbers with ease and fluency.       

Through organized practice and experiences in various forms such as algorithms/procedures this proficiency and fluency in numbersense is translated into numeracy.

Numeracy is a child’s ability and facility in arithmetic operations (addition, subtraction, multiplication, multiplication, and division) on whole numbers correctly, consistently, fluently, in multiple forms of procedures, including the standard algorithm with understanding. By the end of fourth grade, a child should have mastered numeracy.  If a student has deficits in it at the end of fourth grade, he needs intervention.  That intervention should focus on developing, number concept, arithmetic facts, place value and then numeracy. 

D. Dyscalculia and Acquired Dyscalculia

There are several reasons for the incidence of specific mathematics learning difficulties from language related to neuropsychological and cognitive reasons to environmental reasons.  However, understanding the development of numbersense provides a window into children’s arithmetic difficulties, particularly dyscalculia[1]. Most of the difficulties in mathematics, particularly, arithmetic emanate from the difficulties in learning of number related difficulties. 

Dyscalculia is a child’s difficulty in conceptualizing and using number, mastering number relationships, and producing outcomes of number operations. The difficulty with numbersense may be the result of a child’s assets (or lack thereof)—neurological, neuropsychological, and cognitive reasons, and/or environmental factors—poor teaching, poor curriculum, or lower expectations. When these difficulties exist in spite of a child having intact neurological, neuropsychological, or other cognitive factors, then it is purely because of environmental factors and this will be termed as acquired dyscalculia.  

Dyscalculia or acquired dyscalculia, thus are the manifestation of difficulties in the integration of number concept, numbersense, and numeracy. However, just like effective teaching methods for reading can mitigate the impact of dyslexia, similarly, one can have dyscalculia or acquired dyscalculia, but effective and efficient teaching methods can give students skills so that the effects of dyscalculia and acquired dyscalculia (and other specific mathematics difficulties) are minimized or mitigated. 

Dyscalculia is a quantity/number (with some overlap of spatial orientation/ space organization) based disorder, so the intervention and remedial programs should focus on the development of number concept, numbersense, and numeracy. That means, the interventions and remedial instruction for children with learning disabilities (including dyscalculics) initially should focus on the mastery of number concept—visual clustering, decomposition/recomposition, and acquiring the sight facts. 


To design effective methods, understanding the definitions and characteristics of dyscalculia is critical.  For example, children, especially gifted children, may be able to compensate for even massive deficits using one or more of their equally massive strengths.  A child with tremendous memory and fantastic oral comprehension might be able to get around abysmal arithmetic fact fluency for a while to produce an adequate arithmetic result when mathematics is still fairly simple. But if they have deficits in the understanding, fluency and applicability of number concept and numbersense and procedures, they will have difficulty in future mathematics. Similarly, a child with using counting methods may be able to do well on tests and exams, in the early grades, will have difficulty later on.

Unfortunately, school personnel often get this wrong as their own understanding of dyscalculia and arithmetic disability may be limited affecting resources available for such in quantity and quality. This is compounded by the inadequate preparation of special education teachers in mathematics and mathematics learning disabilities, particularly dyscalculia compounding the environmental factors related to dyscalculia. Many parents and school personnel mistakenly assume that dyscalculia equates to permanent condition of deficit or disability. Since, true number concept is at the basis of the development of fluent numbersense, the condition of dyscalculia and acquired dyscalculia can be compounded without proper teaching of number concept.  It is very difficult to master (understanding, fluency and the ability to apply), arithmetic facts and procedures without proper number concept teaching.  


        1.   Catch Them Before They Fall

Early identification and assessment of number concept and numbersense are essential to prevent numeracy failure in young children and avoiding future mathematics difficulties. Students without adequate mastery of sight facts and decomposition/recomposition early, continue to demonstrate poor numbersense and numeracy skills, even into the middle grades and high school. There is a strong predictive validity to early number concept (particularly decomposition/recomposition) and later mathematics achievement. The contribution of sight facts and decomposition/recomposition role does not diminish. Children continue to use the number concept and decomposition/recomposition whenever they encounter numeracy problems. For example, when students encounter work on fractions, integers and rational numbers, they continue to need and use them for new number relationships. Research in reading processes indicates that certain domain-specific deficits such as measures of a child’s phonemic awareness and phonological sensitivity are the best predictors of early reading performance (better than IQ tests, readiness scores, or socioeconomic level). It is widely accepted that deficits in phonological processing are the proximal cause of Reading Disability (RD) (Vellutino, Fletcher, Snowling, & Scanlon, 2004)). 

Similarly, a domain-specific deficit in processing numerosities (numberness) has been implicated in mathematics disabilities (MD), such as dyscalculia  (Butterworth, 2010; Wilson & Dehaene, 2007). And of the several sub-skills of numberness the most important skill involved as a deficit is decomposition/ recomposition.  

In addition, domain-general cognitive risk factors, such as slow processing speed and working memory might be shared between the two disorders and could possibly explain why they may co-occur. Research shows that almost 40% of dyslexic also exhibit symptoms of dyscalculia. This comorbidity between RD and MD indicates that there is fundamental components in both of them that implicate are phonemic awareness and decomposition/ recomposition, respectively. We have observed, with hundreds of children, that fluent numberness (integration of one-to-one correspondence with sequencing, visual clustering, and decomposition/recomposition) is a better predictor of future proficiency and fluency in arithmetic and even higher mathematics. Remediation of numberness related key skills results in better understanding and mastery of numeracy.

On the other hand, high comorbidity rates between reading disorder (RD) and mathematics disorder (MD) indicate that, although the cognitive core deficits underlying these disorders are distinct, additional domain-general risk factors might be shared between the disorders. Three domain-general cognitive abilities processing speed, temporal processing, and working memory are studied in RD and MD. Since attention problems frequently co-occur with learning disorders, the three factors, which are known to be associated with attention problems, account for the comorbidity between these disorders. However, the attention problems observed in the case of MD, some of them are secondary, in the same sense, that they might be the byproduct of consistent failure in mathematics. 

Research on measures of processing speed, temporal processing, and memory with primary school children with RD, children with MD, children with both disorders (RD+MD), and typically developing children (TD controls) show that all three risk factors are associated with poor attention. 

After controlling for attention, associations with RD and MD differed: Although deficits in verbal memory were associated with both RD and MD, reduced processing speed was related to RD, but not MD; and the association with RD was restricted to processing speed for familiar nameable symbols. In contrast, impairments in temporal processing and visuospatial memory were associated with MD, but not RD. Visuospatial memory is essential for visual clustering, decomposition/recomposition, and therefore with development of sight facts.   

2.  Interventions and Remedial Strategies for Dyscalculia

To help LD students, particularly poor readers, become fluent readers, few key components are typically involved: (a) constantly increasing sight vocabulary, (b) sustained, systematic work on phonics and phonological sensitivity as a means to ‘breaking the code’ and build proficiency by focusing on phonemic awareness and (c) repeated readings using efficient strategies for blending sounds to build fluency. The key element is the constant and intense practice in phonemic awareness that helps students to connect graphemes and phonemes. This process helps children to move from decoding of individual letters to chunking and blending sounds. This insight helps interrupt the cycle of failure for poor readers. 

Many planners of mathematics instruction for young children often do not fully take into account that to increase proficiency, competence, and fluency with basic addition and subtraction facts, children need to develop solid number concept (numberness) and flexible numbersense. They stop short: as soon as children are able to count one-to-one, they assume the child has the concept of number. Sequential counting and one-to-one correspondence, even when it is converted into conservation of number is not enough for competence in numbersense.

For example, explicit teaching of phonemic awareness skills and sound blending skills is important, but instruction that integrates ‘in how to blend phonemes together’ and also how to ‘pull apart’ or ‘segment words into phonemes’ is more useful to students in order to acquire reading skills. The organized and intense supervised practice in building sight vocabulary, “pulling apart” and “blend it together” converts novices into fluent readers.  Similarly, learning sequential counting, one-to-one correspondence, visual clustering, building sight facts, and decomposition/recomposition in isolation are useful to an extent, but what is even more important and productive is instruction that focuses on their integration. An organized and early intensive supervised practice in the integration of these component skills develops true number concept (numberness) and then aids in the optimal development of numbersense.  


When one hears or sees a number, one does not see discrete objects; instead, one sees or hears a collection represented in its abstract symbolic form.  Children can count objects one-by-one, but they have difficulty recognizing visual clusters of objects as represented by specific numbers. Counting is like recognizing and decoding individual letters and their sounds, and recognizing visual clusters for individual numbers is like recognizing phonemes and even words. 

Decoding individual letters does not make a child a fluent reader. For proficiency and fluency in reading with comprehension, one needs graphemephoneme connection and practice in blending sounds. Similarly, counting does not make a child fluent in numberness or numbersense.  One has to blend numberness of two numbers (sight and addition facts) to produce new numbers. Arithmetic facts (blending of two numbers) and place value (blending of two or more numbers) are like identifying the phonemes in a big word and then blending of those sounds in reading that word.  Weaknesses in phoneme awareness, rapid automatized naming and working memory are strong and persistent correlates of literacy problems, particularly spelling, even in adults. Similarly, decomposition/recomposition, working memory, and rapid automatized naming are related to addition and subtraction facts and then with multiplication and division facts.  

Strategies and instruction in arithmetic facts and procedures are much more productive when a child has acquired the number concept properly. Further, true number concept and decomposition/recomposition is the basis of deriving addition and subtraction strategies [Making ten, N + 9 (add 10 – 1), N+N (doubles), N + (N+1) (doubles plus 1), N + (N – 1) (doubles minus 1), N + 2 ( 2 more), (N + 1) + (N – 1) (2 apart)]. Mastery of arithmetic facts, thus, is dependent on number concept, decomposition/recomposition, and strategies based on these. For example, 8 + 6 = 8 + 2 + 4 (applying the knowledge that we need 2 to make 10 and 6 is decomposed into 2 + 4) = 10 + 4 (8 and 2 are recomposed into 10 and with the knowledge of teen’s numbers—place value to get 14), or 8 + 6 = (7 + 1) + 6 =  7 + 7 = 14, or 8 + 6 = 2 + 6 + 6 = 2 + 12 = 14, or 8 + 6 = 8 + 8 – 2 = 16 – 2= 14.  All of these strategies are built on sight facts, making ten, and decomposition/recomposition

As phonemic awareness is to reading, decomposition/recomposition is to numbersense and numeracy.  The reading research demonstrates phonemic awareness (PA) is one of the biggest building block of the reading success, decomposition/recomposition process is the building block to mathematics.  And, PA should not be abandoned until the child has demonstrated advanced levels of PA skills. However, decomposition/recomposition, as a process, is present in mathematics at all levels; it is never abandoned. 

A skilled reader is able to read almost every word without activating the phonological processor. Fluent reader reaches that point because she did use the phonological processor which allowed her to make words automatic. There is a real difference between a beginning reader and a fluent reader.  Looking at a cluster of objects (up to ten) and recognizing it and giving it a numerical name, instantly, is the goal, but to get there we need to focus on recognizing smaller clusters and even some counting.  As in reading a word, we do not focus on each letter, nor we begin with the whole word memorization, similarly, in numberness, we neither focus on one object at a time nor on the whole cluster to start with.  How can every child can do it in a

reasonable time frame and with fluency is the goal of next several sections?

Thus, the acquisition of number concept and then acquiring arithmetic facts, at one level, is similar to learning to read: one acquires the meaning of symbols, decodes the symbols and then combines the symbols to relate to other symbols. In reading, it is the process of associating phonemes to grapheme and vice-versa and the blending of sounds. When a child encounters a new word, she breaks the word into “chunks” and then “blends” these chunks into the word. This chunking and blending is the manifestation of the integration of grapheme-phoneme relationship. A child cannot become a fluent reader without being fluent in grapheme-phoneme association and the process of  “chunking” and “blending”. 

The parallel process in mathematics is also three-fold: (a) instantly associate a visual cluster with its number representation, (b) associating a sound with number, and (c) decomposition and recomposition.            

The recognition of a visual cluster and decomposing a cluster into its subclusters and combining sub-clusters into a larger cluster and instantly recognizing the larger cluster as the combination of the sub-clusters are the key processes to acquiring the number concept and numbersense.  In most situations, in teaching number, teachers help children acquire the concept of number by connecting the phoneme (sound s-e-v-e-n) with grapheme (symbol 7). But that is not enough because acquiring numberness is more than learning to read a number or even count. This causes the manifestation of the symptoms of dyscalculia and acquired dyscalculia in most children.  

In teaching letters, there are two schools/traditions: in England, the sound of letters are taught first and then the letters, in most American schools, the letters first and then their sounds. In reading research, there is higher correlation between sound/letter to reading than letter/sound relationship. There is great deal of evidence that both are important. Similarly, in the United States, the number names are taught first and then the numerosity associated with them. In contrast, in Asia and England, it is the numerosity first and then the number shapes. Both are essential for learning number concept.  However, when numerosity and oral representation are learnt before the writing of numbers, children develop number concept faster. The judicious integration of the two expedites the process.


[1] See Dyscalculia by Sharma and Loveless (1988)








Mathematics For All

754 Old Connecticut Path Framingham, MA 01701

877 4089 (T): (508) 494 4608 (C) 

(508) 235 7200 (F)


Center for Teaching/Learning of Mathematics 

Programs and Services


CT/LM has developed programs and materials to assist teachers, parents, therapists, and diagnosticians to help children and adults with their learning difficulties in mathematics. We conduct regular workshops, seminars, and lectures on topics such as: How children learn mathematics, why learning problems occur, diagnosis, and remediation of learning problems in mathematics.

1. How does one learn mathematics?  This workshop focuses on psychology and processes of learning mathematics—concepts, skills,
and procedures. Participants study the role of factors such as:  Cognitive development, language, mathematics learning personality, pre-requisite skills, and conceptual models of learning mathematics.  They learn to understand how key mathematics milestones such as number conceptualization, place value, fractions, integers, algebraic thinking, and spatial sense are achieved.  They learn strategies to teach their students more effectively.

2. What are the nature and causes of learning problems in mathematics?  This workshop focuses on understanding the nature and causes of learning problems in mathematics.  We examine existing research on diagnosis, remedial and instructional techniques dealing with these problems. Participants become familiar with diagnostic and assessment instruments for learning problems in mathematics. They learn strategies for working more effectively with children and adults with learning problems in mathematics.

3. Content workshops.  These workshops are for teachers and parents on teaching mathematics milestone concepts and procedures. For example, they address questions such as:  How to teach arithmetic facts easily?  How to teach fractions to students more effectively? 
How to develop the concepts of algebra easily?
 In these workshops, we use a new approach called Vertical Acceleration. In this approach, we begin with a simple concept from arithmetic and take it to the algebraic level. 
We offer individual diagnosis and tutoring services for children and adults to help them with their mathematics learning difficulties and learning problems, in general, and dyscalculia, in particular. We provide:


1. Consultation with and training for parents and teachers to help their children cope with and overcome their anxieties and difficulties in learning mathematics.

2.  Consultation services to schools and individual classroom teachers to help them evaluate their mathematics programs and help
design new programs or supplement existing ones in order to minimize the incidence of learning problems in mathematics.

3. Assistance for the adult student who is returning to college and has anxiety about his/her mathematics.

4.      Assistance in test preparation (SSAT, SAT, GRE, MCAS, etc.)

5.  Extensive array of mathematics publications to help
teachers and parents to understand how children learn mathematics, why learning
problems occur and how to help them learn mathematics better.


The Math Notebook (TMN)

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

Selected Back Issues of The Math Notebook:

Children’s Understanding of the Concept of Proportion – Part 1 and 2 (double)

A Topical Disease in Mathematics: Mathophobia  (single) Pattern Recognition and Its Application to Math  (double)

Mathematics Problems of the Junior and Senior High School Students  (double)

Mathematically Gifted and Talented Students  (double)

Types of Math Anxiety  (double)

Memory and Mathematics Learning  (double)

Problems in Algebra – Part 1 and Part 2 (special)

Reversal Problems in Mathematics and Their Remediation  (double)

How to Take a Child From Concrete to Abstract  (double)

Levels of Knowing Mathematics  (double)

Division:  How to Teach It  (double)

Soroban: Instruction Through Concrete Learning  (double)

Mathematics Culture  (double)

Mathematics Learning Personality  (double)

Common Causes of Math Anxiety and Some Instructional Strategies  (double)

On Training Teachers and Teaching Math  (double)

Will the Newest “New Math” Get Johnny’s Scores Up?  (double)

Dyslexia, Dyscalculia and Some Remedial Perspectives For Mathematics Learning Problems  (special)

Place Value Concept:  How Children Learn It and How To Teach It  (special)

Cuisenaire Rods and Mathematics Teaching  (special)

Authentic Assessment in Mathematics  (special)

Math Notebook: Single issue ($3.00); Double issue ($5.00); Special issue ($8.00)

FOCUS on Learning Problems in Mathematics

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

Selected back issues of Focus:

Volume 3, Numbers 2 & 3: Educational Psychology and Mathematical Knowledge

Volume 4, Numbers 3 & 4: Fingermath: Pedagogical Implications for Classroom Use

Volume 5, Number 2: Remedial and Instructional Prescriptions for the Learning

Disabled Student in Mathematics

Volume 5, Numbers 3 & 4: Mathematics Learning Problems and Difficulties of the Post

Secondary Students

Volume 6, Number 3: Education of Mathematically Gifted and Talented Children

Volume 6, Number 4: Brain, Mathematics and Learning Disability

Volume 7, Number 1: Learning Achievement:  Implications for Mathematics and Learning Disability

Volume 7, Numbers 3 & 4: Using Errors as Springboards for the Learning of


Volume 8, Numbers 3 & 4: Dyscalculia

Volume 9, Numbers 1 & 2: Computers, Diagnosis and Teaching (Part One and Two)

Volume 11, Numbers 1 & 2: Visualization and Mathematics Education

Volume 11, 3 (1989): Research on Children’s Conceptions of Fractions

Volume 12, Numbers 3 & 4: What Can Mathematics Educators Learn from Second Language Instruction?

Volume 13, Number 1: Students’ Understanding of the Relationship between Fractions and Decimals

Volume 14, Number 1: The Psychological Analysis of Multiple Procedures

Volume 15, Numbers 2 & 3: Vygotskian Psychology and Mathematics Education

Volume 17, Number 2: Perspective on Mathematics for Students with Disabilities

Volume 18, Numbers 1-3: Gender and Mathematics:  Multiple Voices

Volume 18, Number 4: The Challenge of Russian Mathematics Education:  Does It Still Exist?

Volume 19, Number 1: Components of Imagery and Mathematical Understanding

Volume 19, Number 2: Problem-Solution Relationship Instruction: A Method for

Enhancing Students’ Comprehension of Word Problems

Volume 19, Number 3: Clinical Assessment in Mathematics: Learning the Craft

Volume 20, Numbers 2 & 3: Elements of Geometry in the Learning of Mathematics

Volume 22, Numbers 3 & 4: Using Technology for the Teaching and Learning of Mathematics


Volume 23, Numbers 2 & 3: Language Issues in the Learning of Mathematics Volume 28, Number 3 & 4: Concept Mapping in Mathematics

Professor Mahesh Sharma is the founder and President of the Center for

Teaching/Learning of Mathematics, Inc. of Framingham, Massachusetts and Berkshire Mathematics in England. Berkshire Mathematics facilitates his work in the UK and Europe.

He is the former President and Professor of Mathematics Education at Cambridge College where he taught mathematics and mathematics education for more than thirty-five years to undergraduate and graduate students. He is internationally known for his groundbreaking work in mathematics learning problems and education, particularly dyscalculia and other specific learning disabilities in mathematics. 

He is an author, teacher and teacher-trainer, researcher, consultant to public and private schools, as well as a public lecturer. He was the Chief Editor and Publisher of Focus on Learning Problems in Mathematics, an international, interdisciplinary research mathematics journal with readership in more than 90 countries, and the Editor of The Math Notebook, a practical source of information for parents and teachers devoted to improving teaching and learning for all children.

He provides direct services of evaluation and tutoring for children as well as adults who have learning disabilities such as dyscalculia or face difficulties in learning mathematics. Professor Sharma works with teachers and school administrators to design strategies to improve mathematics curriculum and instruction for all.

He regularly offers workshops on mathematics education for professional development at Framingham State University.  For registration, please call Anne Miller (508 215 5837), or visit: professionaldevelopment/



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