My experience in introducing young children, with or without learning disabilities, to number concepts and number relationships shows that language alone (without any or limited concrete manipulative experiences) cannot effectively develop the concept of number. Some initial understanding of number is independent of language. For example, young children acquire the concept of number earlier than they do pluralization rules, showing that they can learn number one and generate other numbers from one. This is achieved at a concrete level. However, concrete experiences alone are not enough for the abstraction of number from specific concrete activities to a general notion of numberness and number sense. For that they need language, rich language.
For every concrete experience to reach a conceptual level and to last as a concept that can be communicated, an appropriate and stable label (language container) is needed for the concept being abstracted and acquired from the concrete experience.
Concrete experiences accompanied with language (questions being asked, commentary being made, descriptions being furnished) convert the concrete experiences into abstractions with labels. These interactions between concrete experience and language provide such labeling (language containers) and abstractions inherent in them. Persons in the child’s environment (parents, teachers, and siblings) mediate this interaction in the form of posing questions, affirming responses, and creating practical problem solving situations.
Conceptualization of number begins with concrete experiences, but its acquisition is facilitated and accelerated by language and appropriate concrete materials and actions. Scaffolding questions accelerate the symbolizing and abstracting from concrete experiences.
In other words, concrete experience is the starting point for number conceptualization, and language furthers the development of the abstraction process. In most cases, language experiences alone are not enough for number conceptualization. In the initial states, language plays a role, but it is a limited one. Initial number conceptualization is not an artifact of language because it is dependent on concrete models. The interaction of rich language and quantification with efficient concrete materials facilitates the development of number concept. Once the number is conceptualized and there are adequate language containers, in most cases, further concepts can be developed with language alone. However, it is not advisable as the purpose of concrete materials is to help to
(a) generate the language (language containers—vocabulary),
(b) create conceptual schemas, and then, finally,
(c) arrive at symbolic/procedural representation of the concept.
The role of concrete materials is not to solve every problem with them or to become dependent on them as we see in many special education settings. To gain efficiency and generalization, children should transcend the concrete materials and language containers and should be able to model practical problems using mathematical ideas and schemas. Ultimately, children should express their understanding and problem solving ability through mathematical symbols and their manipulations such as formulas, expressions, and equations.
Mathematics Language and Native Language
Once number is conceptualized, its applications and related concepts are much more dependent on language. Number conceptualization is the beginning of the development of the language of mathematics—its terms, words, order of usage, and translations from mathematical expressions to English language and vice versa. Mathematics is a unique language. Numbers are its alphabet and number symbols are its smallest thinking units. Mathematical symbols, concepts, and language modify the natural language, and they in turn shape mathematical thinking.
In the child’s native language system and in the context of concrete experiences, numbers and quantification function as predicators and qualifiers—modifying the nouns in their scope and meaning quantitatively, such as five dishes, two toys, many books, fewer children, some books, a pen, etc. They function, in some respects, like adjectives in a sentence in the native language. Children are quite fluent in this before they begin school as pre-Kindergartners or Kindergartners.
Later, in the language of mathematics, numbers are real and even concrete and do not generally function as predicators; number characteristics are described by other predicators. For example, six hundred is a big number, an even number, or a much smaller than the number six hundred thousand. Thus, in the formal language of mathematics, numbers are abstract, singular objects, but at the same time, in mathematical operations, numbers are ‘real,’ concrete entities—for example, when a child says: “I will show you number four” and then shows four fingers. This thinking of numbers as concrete and abstract at the same time is at the core of true number conceptualization.
Holding in the mind the idea that a number is concrete and abstract at the same time is difficult for many children. At the same time, this need to understand numbers as concrete and abstract creates hurdles for many children to become fluent in its usage in other mathematics concepts and its applications outside of mathematics. Thinking of number as concrete and abstract at the same time is therefore a real challenge for many children. It is perhaps the most paradoxical thing about numbers—their concreteness and their abstract representation, and this is the feature that makes them so useful.
Transitions: Concrete to Abstract and Native to Formal Language
Acquisition of number concepts, then, requires that the child perform two types of abstractions: one abstraction when a child translates sensory, concrete representations of quantity into symbolic entities (five fingers is translated into the number 5). The second transition is when the child transforms the conception of number as a predicate in his native language to its conception as an object in the language of mathematics (I am showing you five fingers to look this is 5.). For some children, particularly learning disabled (LD) children, these transitions are not easy and need to be facilitated carefully by a knowledgeable person using appropriate language, an enabling questioning process, and efficient instructional models.
These two transitions are significant and involve complex processes. They evolve through a prolonged constructive process, which begins with native language and concrete experiences such as counting on fingers and ends in the child’s conception of number as part of the formal mathematics language and symbolic logical system. In the elementary grades, the child is constantly struggling with these roles of number and the interrelationships between the predicative function of numbers in the native language and their status as objects in the language of mathematics.
The mediation between the native language and the language of mathematics depends on the quality of concrete models and linguistic experiences provided to the child. Effective teachers provide a variety of concrete learning experiences, and their expert questioning and deft use of language facilitate these transitions.