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.