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.)

❒   ❒   ❒   ❒     ❒     ❒   ❒

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

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