K-12 Teaching and Learning From the UNC School of Education

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"There were thirteen pencils in our classroom pencil box. Six kids each took a pencil. How many pencils were left?"

This problem is posed in a second-grade classroom. Children are working in groups to figure out how to solve it using strategies that work for them:

  • One group counts out thirteen cubes, takes away six of them, and counts the remaining seven cubes.
  • In another group, one child draws thirteen marks on a piece of paper, another child crosses out six marks, and another child counts the seven marks that are left.
  • Another group uses a "double" to reason that 12 minus 6 is 6, so 13 minus 6 is one more than that, or 7.
  • In another group, children are discussing how this could be related to a "ten" fact that they already know: "13 take away 3 is 10, take away 3 more to get 7."
  • In still another group we hear someone counting out loud, "13… 12, 11, 10, 9, 8, 7" and someone else is holding up a finger each time the next number is said until she has six fingers showing.

All of these groups of children are using strategies that reveal the different ways the children are thinking mathematically as well as their flexibility with numbers. This intuitive feel for numbers and their relationships can be called number sense, an important component of the elementary mathematics curriculum

What is number sense?

Number sense "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 skill in working with them."1 Number sense is not a discrete set of skills to be taught for three weeks in October or something that only those that are "good at math" have. It is a part of children’s daily mathematical lives and slowly grows and develops over time. In a problem-centered mathematics curriculum, number sense is closely tied to problem solving, as the children described above show. These children have learned, over time, that they are capable of solving problems and that they can play with numbers to make sense of a problem. They have used their growing number sense to develop strategies to help them solve problems.

What types of strategies do children develop?

As children become problem solvers, certain types of problem-solving strategies often emerge in their work. Trafton and Thiessen, in Learning Through Problems: Number Sense and Computational Strategies, describe some strategies that are commonly seen among children in the primary grades. They give examples of how a child might use these strategies to solve the equation 29 plus 14:

  • Partitioning numbers using tens and ones. "First I added the 20 and 10 and got 30. Then I added the 9 and 4 and got 13. Then I added the 10 from 13 to 30 and added 3 more and got 43.
  • Counting on or back from a number. "First I counted on from 29 by tens and went 29, 39. Then I counted on 4 more — 40, 41, 42, 43."
  • Using "nice numbers." Nice numbers are multiples of 10 or other numbers that are easy to work with. "I know that 30 plus 15 is 45, but 29 plus 14 is 2 less than that, so it’s 43."
  • Translating to a new problem. "I took one away from the 14 and gave it to the 29 to make 30. Then I had 30 plus 13, which is 30 plus 10 plus 3, which is 43.

By developing these strategies themselves, using them to construct meaning in problems, and listening to other children describe their use of these strategies, children’s facility with numbers and confidence about their ability to do mathematics increase

In what other ways can number sense develop?

In addition to group work with problem solving, number sense can be developed in the context of a variety of classroom activities such as use of graphic representations, daily routines, and games. Below are some activities that can help your students become comfortable with numbers.

Graphic representations

ten frame with 6 counters

Organizing numbers and representations of numbers graphically can help children move quickly and flexibly with counting and computation. A ten frame (shown at right), found in a variety of text and resource books, helps children work easily with 10, a "nice number." Briefly show children a ten frame with, for example, 6 counters in the spaces, and then ask them how many counters there are and how many spaces without counters there are. By repeating this activity frequently with other numbers of counters from 1 to 10, children become adept at recognizing pairs of numbers that partition 10 and at using 5 and 10 as benchmarks.

A hundreds chart is another valuable tool that contributes to children’s number sense, especially when introduced early in the school year and used throughout the year. Building a hundreds chart helps children examine the relationships among the numbers. Starting with any number and moving up or down a column allows children to count by 10. Children can also practice counting by 2, 3, 4, and so on. Given a hundreds chart with only a few numbers filled in, children can fill in the rest of the chart by recognizing number relationships. Children can also use a completed hundreds chart as a visual support for problem solving strategies. While some children may need to count by ones on the chart, with time and multiple experiences, they are likely to develop an understanding of the "shortcuts" that the rows and columns provide.

Daily routines

Many teachers use the calendar as a source of mathematics activities. Children can work with counting, patterns, number sequence, odd and even numbers, and multiples of a number; they can also create word problems related to the calendar. A hundreds chart can help them count the number of days in school, and the current day’s number can be the "number of the day." Children can suggest various ways to make or describe that number. For example, on the 37th day of school, children may describe that number as 30 plus 7, 40 minus 3, an odd number, 15 plus 15 plus 7, my mother’s age, or 1 more than 3 dozen. The complexity of children’s responses will grow as the year goes on and as they listen to one another think mathematically.

Games

Games can be used in the context of a lesson or be made available for children to play with numbers throughout the day. The calculator can be a fun tool for quick mental computation practice. To make a "Calculator Two-More-Than Machine" (Van de Walle, 2001), press 0 + 2 = on the calculator. Then press any other number, such as 7, and hold your finger over the = key. Have the children predict the sum then press the = key to confirm it. As long as they continue to press only a number and =, the "machine" will continue to add 2. The use of the calculator and the immediate feedback reinforce computation and encourage children to keep playing.

A simple game to play when you have a few minutes to fill is "Guess My Number" (Bresser & Holtzman, 1999). Choose a secret number and tell children a range that your number falls within. You can start small, with 1 to 10, or use a larger range (such as 1 to 100, 25 to 75, or 150 to 250). Have children guess your number and tell them whether their guess is larger or smaller than your number. Children will quickly develop strategies that help them zero in on your secret number. To extend this game, choose a secret number from a wide range such as 1 to 500 and give one clue, such as that it is even, it ends in 4, or the sum of the digits is 9; then ask students to start guessing.

To play "Stand Up and Be Counted" (Bresser & Holtzman, 1999), first ask children to describe the number 25 in as many ways as they can, as with the "number of the day," and record their ideas as an example. Then have each child draw from a bag of squares numbered 1 through 100 and write down as many ways as they can to make the number they drew. Ask a volunteer to stand up and read one statement at a time about his or her number. If that statement is true for other children’s numbers, they stand up. If it is not true, they remain seated. Through discussion, the children can begin to focus on the characteristics of the numbers and their relationships

Number sense every day

All of these number sense activities contribute to your students’ abilities to solve problems. When children have daily, long-term opportunities to work (and play) with numbers, you will be continually amazed by the growth in their mathematical thinking, confidence, and enthusiasm about mathematics. By helping your children develop number sense, especially in the context of problem solving, you are helping them believe in themselves as mathematicians

References

Bresser, R. and Holtzman, C. (1999). Developing number sense: Grades 3-6. Sausalito, Calif.: Math Solutions Publications.

Trafton, P.R. & Thiessen, D. (1999). Learning through problems: Number sense and computational strategies. Portsmouth, N.H.: Heinemann.

Van de Walle, J. A. (2001). Elementary and middle school mathematics: Teaching developmentally. 4th ed. New York: Addison Wesley Longman.

Notes

1 Trafton and Thiessen (1999), p. 50.