# Understanding a Child's Development of Number Sense

Number sense refers to a person's general understanding of number and operations along with the ability to use this understanding in flexible ways to make mathematical judgments and to develop useful strategies for solving complex problems (Burton, 1993; Reys, 1991). Researchers note that number sense develops gradually, and varies as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms (Howden, 1989). Developing students' understanding of numbers, ways of representing numbers, relationships among numbers, and number systems are focus areas for grades PreK–2.

In the primary grades, children move from developing basic counting techniques to understanding number size and relationships, place value and operations. However, students develop this understanding at different rates. Some young children can count 13 objects, but they do not see that 13 is one group of ten and 3 ones. Others can group objects into tens and ones and understand their numerical representation.

The video clips in this example illustrate the range in understanding of numbers and their relationships of students in grades Pre K-2. Cena and Jonathan age 7, and Rudy, age 9, demonstrate different levels of understanding numbers and place value concepts.

### Beginning Understanding of Place Value and Number Representation

In the first video clip below, Cena explains how she would group objects in tens and ones during a whole-class discussion. What can you say about her understanding of numbers after listening to her explanation?

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In this video clip, Cena is asked by another adult to show the value of each of the digits in 18 chips. This interviewer was a familiar face to the children, having worked with the class previously.

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In this first video example, Cena demonstrates a partial understanding of place value, but she has not yet developed an understanding of the tens and ones structure and the relationship to standard symbolism of numerals. In the whole class setting, she shows understanding of grouping ten 1's to make 10. However, during the individual interview, when she was asked to show the value of the 1 digit in 18 using chips, she shows 1 chip, not 10.

 Discussion Questions Why might Cena be able to write two-digit numerals correctly without being able to explain the meaning of digits? What reasons might there be for the differences in Cena's performance during the whole class lesson as compared to the individual interview?

### Understanding Grouping and the Value of 10

In the video clip below, Jonathan is telling the teacher how he would group a set of items into 10's and 1's. Notice the confidence Jonathan shows in his response.

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In this second video clip, Jonathan is asked to explain his understanding of place value concepts to an interviewer. Notice the hesitancy in his responses.

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In this second example, Jonathan illustrates a partial yet developing understanding of numbers and their symbolic representation. He does display an understanding that the 1 in the number sixteen equals 10 chips. However, when trying to explain the meaning of 16, he has difficulty articulating his ideas without help.

 Discussion Question How would you describe Jonathan's understanding of place value as compared to Cena's understanding?

### Developed Understanding of Number Sense and Place Value

In this third video example, Rudy demonstrates a developed sense of numbers and a strong understanding of place value. While watching this video clip, consider how much effect you think childrens' ages and developmental levels have on their ability to understand place value.

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 Discussion Question How would you compare Rudy's understanding of place value to Cena's and Jonathan's understandings?

### Conclusion, Reflection and Discussion

• Do you think the children would have done better if the place value interview question had used numbers above twenty instead of numbers in the teens? In English, teen numbers are especially difficult. Compare Asian languages, in which 13 is pronounced, "one ten and three," to the English word "thirteen." It is not clear how "three" and "thirteen" connect.
• Do you think it was "fair" that the children were not all asked the same questions in the same way? Can we fully probe a young child's understanding without the type of freedom that a clinical interview such as these provides?
• Interviewers must be sensitive to how children interpret requests such as "represent the '1'." Might a child be confused unless this is carefully crafted to ensure the child knows the interviewer means "the part of the initial teen number that the '1' represents"?
• Would you agree that basing an assessment of a child's understanding with just standard curriculum tasks is inadequate? How might such tasks be altered so that teachers can observe individual children in the context of classroom work?

### References

Burton, G. M. (1993). Number sense and operations: Addenda series, grades K-6. Reston, VA: National Council of Teachers of Mathematics.

Howden, H. (1989). Teaching Number Sense. Arithmetic Teacher 36(6): 6-11.

Reys, B. J. (1991). Developing number sense: Addenda series, grades 5-8. Reston, VA: National Council of Teachers of Mathematics.

Videoclips used by permission from Mathematics: Assessing Understanding. Part 1 by Marilyn Burns, ©1993 by Cuisenaire Company of America, an imprint of Pearson Learning.

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