Mastering the Concept of Number: Numbersense

Just like language, the awareness of quantity and space is socially mediated. Culturally expected achievement related to quantity and spatial skills initiates and supports the awareness and mastery of number related skills. The cultural tools – language, games, toys, social interaction, and related goals and expectations – maximize the formation of the idea of number and its usage. Acquiring number concept and the usage of numbers is a cultural tool in most advanced cultures and the first of a series of tools for being productive as a citizen, solving survival problems, and demonstrating cognitive potential—mathematical ways of thinking.

Just like learning the use of any tool, the outcomes can be enhanced if we learn how to use these tools efficiently, appropriately, and effectively. For this reason, the introduction to and use of these tools by children should begin as early as possible.

Apart from language, calculation is perhaps the only culturally determined system in the modern world that the majority of the population is expected to master. For success in today’s technological world, it is important to have good number and spatial sense. In our world, one needs to be literate and numerate.

Being numerate means a person has flexibility with the use of numbers. It means having a good sense of number and number relationships. And it means an ability to make use of number skills, which enables an individual to cope with the practical quantitative demands made by everyday life, for instance the numerical trends in graphs, charts, or tables, or in reference to percentage increase or decrease. Numerates—practitioners of numeracy have advantage in the modern world.

Numeracy, like literacy, is a complex phenomenon. Numeracy is the demonstration of proficiency in various number related skills. Numeracy is the ability to execute four whole numbers operations, in the standard form, correctly, consistently, efficiently, and with understanding. To be fluent in numeracy includes having a good numbersense. Flexibility in handling quantity is called numbersense. Numbersense refers to a person’s ability to look at the world quantitatively and make quantitative and spatial comparisons and decisions using mental calculations. Technically, it is the integration of (a) number concept, (b) number relationships (arithmetic facts), and (c) place value. With practice and experience, this proficiency and fluency in numbersense translates into numeracy.

 Numbersense describes a cluster of ideas such as the meaning of a number, 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).

Numbersense is not a set of discrete skills but a set of integrative skills. Students with good number sense 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, individuals with a good sense of number can decompose and recompose numbers with ease.

Understanding the concept of numbersense provides a window into children’s arithmetic difficulties, particularly dyscalculia. Dyscalculia is a child’s difficulty in conceptualizing number, mastering number relationships, and producing outcomes of number operations.

The difficulty may be the result of a child’s assets—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 assets, then they are purely because of environmental factors – the term for such difficulties is acquired dyscalculia.

Dyscalculia or acquired dyscalculia results in the manifestation of difficulties in the integration of number concept, numbersense, and numeracy. But just as most dyslexics can learn to read with efficient teaching methods, in most cases, those with acquired dyscalculia and even dyscalculia can learn mathematics with effective and efficient strategies. Thus, one can have dyscalculia or acquired dyscalculia, but effective and efficient teaching can give skills so that the effect of dyscalculia or dyscalculia is mitigated. A person would still have dyscalculia, but he/she will not be disabled.

More current definitions of dyscalculia are critical. Children, especially gifted children, may be able to compensate for even considerable deficits using one or more of their equally substantial strengths. For a while, children with tremendous memory and oral comprehension might be able to cope with lack of arithmetic fact fluency to produce adequate arithmetic results when mathematics is still fairly simple. But if they have deficits in the understanding, fluency and applicability of number concept and numbersense and procedures, then they are disabled for future mathematics. Unfortunately, school officials and parents often have an insufficient understanding of the connection between dyscalculia and arithmetic. As a result, they may assume that all conditions related to dyscalculia equate to disability.

True number concept is at the basis of the development of fluent numbersense. It is difficult to master (understanding, fluency and the ability to apply) arithmetic facts without proper number concept. And, numbersense—number relationships, arithmetic facts, and place value, is necessary for number work, procedures and meaningful problem solving.

Every child should acquire a sense of what numbers represent and be fluent with arithmetic facts such as addition and subtraction, number relationships, multiplication tables, and division facts. Every child should be able to use what he knows to calculate accurately and efficiently, both mentally and on paper, by taking advantage of a range of calculation strategies. These skills are necessary for building a solid foundation for numeracy. The acquisition of these skills is dependent upon appropriate teaching and learning experiences.      

Number Conceptualization: Numberness/Numerosity
To most people, knowledge and use of the first nine natural numbers (one, two, three… up to nine) 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. Most children, in the natural course of living, are able to progress from working with objects to representing these experiences in pictures and icons, to representing them in abstract symbols, and then to manipulating those numerical symbols.

Numbers can be represented in three main formats: Hindu-Arabic (numbers in numerical format), verbal (graphemic or phonological word format), and magnitude-related. Learning number is based on a functional relationship 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 or quantity. Just counting objects is not a number concept. For his reason, we want to introduce another representation of quantity in the form of visual clusters. Visual cluster representation subsumes and extends the subitizing, on the one hand, and the magnitude of number, on the other. Thus, understanding of number is the integration of (a) Hindu-Arabic representation (grapheme, (b) verbal (phonemic) and (c) visual cluster of the number.

An average child 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 those who are not secure in their ability to read and write numerals and to visualize sets of objects or who are not secure in their sense of number and their applications.

If environmental factors are ruled out, difficulty in acquiring numbersense appears to be related to particular deficits in the learning mechanism and specific learning disabilities. Fortunately, early instruction involving activities that develop numbersense can limit and even prevent failure in numeracy and in later mathematics. It is therefore important to teach the integration of numbersense activities with a focus on “sight number fact” automaticity. As the acquisition of sight vocabulary plays a big role in early reading, similarly, the acquisition of sight facts plays a significant role in early numbersense mastery.

Mastering the Concept of Number: Numbersense

Mastering the Concept of Number

Lexical Entries: Number Names
Most children come to school able to recite the string of alphabets as a song—a rhyme. Similarly, many children begin counting by rote—the recitation of an ordered sequence of responses using concrete objects, long before school, particularly in homes where there is emphasis on learning letters and numbers.

In many cases, this rote counting does not mean real understanding of number or its effective usage. Children acquire this skill by observing others using a series of number words in their day-to-day living. Exposure to other people counting or a child’s counting adds to the development of a larger vocabulary for number names—a prerequisite for number concept. It is somewhat similar to a child acquiring sight vocabulary words. Just as 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 the numbers up to about 30.

Counting by rote is an indication of the presence of lexical entries relating to number (number names). Though knowing number names (these lexical entries) is essential, the recitation of numbers is 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 five-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.)

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Child: (Child sequentially counts by touching each cube once) Seven.
Teacher: When you were counting, what number came just before seven?
Child: I don’t 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. Still, this rote counting process is essential as it develops the lexical entries for numbers. 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. This rote counting is a starting point for number conceptualization.

To extend the lexical entries for numbers beyond the first few, 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. Regular rote counting is like reading to young children (even as young as two years old). When we read to very young children, they are acquiring the structure of the language—the intonation, pronunciation, and sound—orthographic symbol correspondence. They are also becoming familiar with the affect associated with words/language. Similarly, when we count with children, they become familiar with the sequence, the structure, and the ability to differentiate the number words and non-number words. This process of regular and frequent counting activates the imaginary number line in children’s minds, just like the exposure to rich language activates the grammar and structure of the language of that culture. At the end of Kindergarten, children should be able to count, forward and backward, by 1, 2, and by 10 from any number at least up to 100.

From Egocentric Counting to the Cardinality of the Set
To be meaningful, counting behavior must have the underlying cognitive structures and processes and the support of language. Many children of ages 4 through 6 can count objects in a rote manner where, for them, the last number uttered represents the “cardinality of the set”; however, they may not know what that means. 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 quantity representing that collection—a property of the collection (“These are six cubes.” “There are six cubes here.”). The transition from simple counting to an understanding of the number as the property of the collection 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?
Child: With a little hesitation he raises his right hand.
Teacher: Good!
Teacher: Can you tell me how many cubes there are on the table? (Points to a collection on the table.)

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Child: (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. I told you there are seven. I counted them again. There are seven. Right?
Teacher: Yes! You are right. There are seven cubes. Let me ask you something. 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 don’t know. Let me try.
The child thinks and counts.
Child: It’s seven.
Teacher: That’s 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. You know what. 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.

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. The child thinks he has produced that number by counting. Assigning a number to the collection always by counting only and even change the number based on recounting is called egocentric counting. The number produced is the product 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 and usage of this count 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. Therefore, children’s use of number requires a true understanding of number, not just rote counting. Associating a number to a collection and considering that number to be the property of the collection is the first step in conceptualizing number properly.

The types of questions asked in these two activities are the means to converting a child’s concrete experiences (egocentric counting) into abstract number conceptualization (number represents the cardinality of the set—it is the property of the set, not of the counting process). The above examples illustrate how the structure of a problem and nature of our questions influence the strategies and concepts children develop to solve problems.

The distinction between the two ways of learning mathematical concepts (drills vs. understanding relations) has a long history in the pedagogy of mathematics learning and teaching. The debate is whether individuals can learn the same mathematical skill either as a set of discrete, non-related rote activities and simple rules or whether they learn concepts and higher order rules with understanding—knowing the interrelationships of component concepts. This artificial distinction also exists in the learning and teaching of number conceptualization. Although the two approaches may result in similar outcomes on tests of isolated skills and the same level of performance on an immediate mastery test, learning with understanding generates broader and deeper learning outcomes with superior transfer to related concepts and longer retention of learned material.

Mastery, in the second example, is not temporary and is not problem and format specific. For example, when a child can derive the sum: 7 + 3 = 10 by counting on fingers, he may not be able to extend it and even remember it from one setting to the other. Counting strategies do not develop “sight number facts” as the results are derived by counting the number each time children want the outcome. When the results are derived by rote counting, there is no visual representation of the number relationships. In contrast, when the child knows 7 + 3 by visualizing it (for example, the display of 10 hearts as a pattern on a visual cluster card and decomposing it as two clusters—one of seven and the other of three hearts), with understanding and automatized mastery – fluency with understanding), then she can extend that 7 of something plus 3 of the same thing is 10 of the same thing.

We have known for a long time that skilled readers are able to read almost every word without activating the same phonological processor they used when first starting to read. They reached that point because they practiced that phonological processor, which allowed them to make words automatic. Reading a word as a sight word is not a strategy, it is a goal. So, how do we get there? Sight words are acquired by continuous and multiple exposures. Likewise, looking at a collection of objects (a cluster of objects up to ten), recognizing it, and instantly giving it a numerical name is the goal, but to get there we need to focus on smaller clusters and even some counting. In reading a word, we do not focus on each letter, nor do 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.

Mastering the Concept of Number