Developing young children's multidigit number sense.

Large numbers are a source of fascination for many children and mathematicians alike. However, unlike mathematicians, most young children have a limited understanding of large numbers and use place value terms indiscriminately to express the enormity of a quantity or measure ("There were thousands of people at the party;.... The house cost millions of dollars." Traditionally, large numbers have not been part of the mathematics curriculum in the early school years. However, a lack of understanding of large numbers can be problematic for young gifted children because large numbers are an integral part of topics that are of interest to them, such as space travel.

In our work with enrichment classes of 5- to 8-year-olds, we found that children were hampered in their investigation of space travel when large numbers were encountered in resource material. Even though these young gifted children were more mathematically competent than their chronological peers in their regular classrooms, they had difficulty in appreciating the number of people who watched the first moon landing; the size of the space mission team; the cost of a space mission; and the distances from the earth to the moon, the planets, and the stars. When children lack an understanding of large numbers, they are unable to reason effectively with the information given. For example, one child reasoned that the moon must be closer than a city because "You can see the moon at night but you can't see Sydney." Thus, for children to take advantage of the information in the space resource material, there was a need to develop their number sense with large numbers, that is, their multidigit number sense.

   Multidigit number sense refers to: Children's understanding of and
   flexibility in using multiunit numbers should also include intuitive
   feelings for numbers and their uses as well as the ability to make
   judgements about the reasonableness of multidigit numbers in diverse
   problem situations (Jones, Thornton, & Putt, 1994, p. 118).

Because multidigit number sense is complex (Jones et al., 1994), it was necessary to develop a series of meaningful activities that enabled young gifted children to make sense of large numbers in context. Enrichment classes provide the opportunity for gifted students to engage in tasks that are beyond the scope of the regular curriculum (Lupkowski-Shoplik & Assouline, 1994). The following enrichment activities were designed to help a class of 20 young gifted children to:

* read (i.e., label) large numbers in symbolic form;

* develop referents for large numbers and understand their relative magnitude (National Council of Teachers of Mathematics, 1998); and

* understand large numbers that represent quantity (e.g., size of a space mission team), distance (e.g., distance to the moon) and money (e.g., cost of a space mission).

Reading Large Numbers

In the first activity, children were introduced to the pattern in reading large numbers. Numbers of increasing magnitude were displayed for the children. We began with the one's period, progressed to the thousand's period, and finally, displayed the million's period. The name of each period was added to facilitate children's reading.

The children enjoyed reading these numbers and were keen to read further in the millions period. Their responses indicated that they had developed the basic skill of reading large numbers: "I learnt that if you try, you can not only count to, but also read large numbers" (Karen, 8 years). Furthermore, even the youngest children demonstrated a primitive knowledge of the structure of our number system as can be seen in 6-year-old Mark's recording of this activity (Refer to Figure 1).

[FIGURE 1 OMITTED]

Following this, the children progressed to three problem-based activities designed to develop the second and third aims, which focused on children's understanding of large numbers in relation to quantity, distance, and money. The children rotated through these activities in small groups.

Understanding Large Numbers

Activity 1: Estimating a Large Quantity

How many peas did Frank knock off his plate?

The book "Counting on Frank" (Clement, 1990) was read to the group and the children were challenged to determine how many peas Frank knocked off his plate (Refer to Figure 2). Initially, the children gave answers such as "a hundred," "a thousand," and "a million" but were unable to agree or explain their estimates. When asked how to find out which answer was correct, their only suggestion was to "count the peas." Other children rejected this suggestion because "It would take too long" and "You can't see all the peas." To provide a basis for solving this problem, the children explored numbers to a thousand and then to one million.

[FIGURE 2 OMITTED]

Exploring Numbers to a Thousand

Referents for 1, 10, 100 and 1,000 were developed using colored sprinkles (confectionery decoration) on buttered bread that was cut into four pieces. The children added sprinkles as follows: 1 sprinkle on the first piece of bread; 10 sprinkles on the second piece; approximately 100 sprinkles on the third piece (by estimating groups of 10); and approximately 1,000 sprinkles on the final piece (by estimating groups of 100).

The sprinkles activity provided a meaningful referent for children's understanding of the relative magnitude of numbers to a thousand, as shown in eight-year-old David's recording (See Figure 3).

[FIGURE 3 OMITTED]

Some children extrapolated beyond the physical referent, making comments such as, "I learnt that you probably can't fit one million sprinkles on one piece of bread" (Helen, 8 years).

We then revisited the original problem about the number of peas Frank knocked off his plate. Although the children were confident that there were more than 1,000 peas, they were unable to suggest an approximate number. We then put the problem aside and explored numbers to a million.

Exploring Numbers to a Million

Multibase arithmetic blocks [MAB] were used to provide a physical referent for one million. The children first explored the area of the meter square using MAB ones, tens and hundreds. They then investigated the capacity of a meter cube using MAB hundreds and thousands. Eight-year-old Aidan explained this process: "We got 12 one-meter rulers, and hundreds of MAB blocks and thousands of MAB blocks. We multiplied and knew how many MAB blocks [would fill the meter cube]."

With some appreciation of numbers to a million, the children were now able to attempt Frank's problem. They reasoned that, because the size of a single MAB was comparable to a pea, Frank had knocked off between two million and five million peas. Although incorrect, their response did follow logically from the pictorial representation of the quantity of peas. It was clear that these large number experiences had informed the children's reasoning in working this problem. For example, 8-year-old Helen explained, "There would be about two million (peas) because the peas are higher than the meter cube and you can't see all the peas in the picture."

Activity 2: Appreciating Monetary Value

What sized container would be needed to carry a million dollars?

In preparation for solving the focus problem above, the children completed two tasks: Money Posters and Monopoly Money. These tasks are described prior to addressing the children's responses to the focus problem.

Money Posters

Posters were labelled with the amounts $1, $10, $100, $1,000, $10,000, $100,000 and $1,000,000. The children identified items in magazine advertisements that approximately cost each of these amounts. They then completed the posters by gluing items under the corresponding amounts. It was evident that this activity raised children's awareness of the monetary value of expensive items, as can be seen in 7-year-old Sondra's comment. "I found out that money could go very large and that you could buy things with a thousand dollars."

Monopoly Money

The children were then given the open-ended task of calculating how much money was in a Monopoly game. Although calculators were provided, the children did not necessarily use them efficiently. For example, one pair of children tediously entered the value of each note in turn, without considering different ways of grouping the notes. In contrast, some children who chose to work manually displayed increasing sophistication in their calculations. This can be seen in Karen's work (See Figure 4).

[FIGURE 4 OMITTED]

After the children had completed these preparatory tasks, they then tackled the focus problem of determining the container size needed to hold a million dollars. The children used the Monopoly money as a referent for working this problem. No containers were provided, rather, the children were encouraged to model different container sizes with their hands. Through discussion, the children realized that there was more than one answer to the problem.

They reasoned that the size of the container was dependent on the denomination of the notes that were used to make one million dollars. Some children commented that a larger sized container would be required if notes of low value were used and vice versa. In other words, the size of the container would be inversely proportional to the denomination of the note that was used. In their search for the smallest container possible, the children's discussion extended to the highest denomination of notes available in other countries.

Activity 3: Stars and Light Years

How far away are the brightest stars?

The purpose of this activity was to develop children's understanding of large distances within the context of space travel. Prior to exploring this problem, the book "How Much is a Million?" (Schwartz, 1985) was read and discussed. The children then made 10 paper stars, which were labelled with the names of the 10 brightest stars (in the Southern Hemisphere), their brightness, and their distance from the earth. The stars were fastened onto upturned paper cups for ease of mobility. The children initially ordered the stars by brightness beginning with the brightest star.

Next, consideration was given to the stars' distances from the earth. After the children had discussed the notion of measuring stellar distances in light years, they reordered the stars from the closest to the most distant. The children spontaneously debated whether there was a correspondence between the brightness of a star and its distance from earth.

To represent the stars' relative distances from the earth in light years, a timeline was drawn and marked in 100's from 0 to 1,000. The children positioned each star at the correct number of light years from the earth. They then discussed the idea that when we see a star today, the light from that star was actually emitted a number of years ago. Fascinated with this idea, the children began working out the year when light was emitted from particular stars and related these years to significant events.

This activity enabled children to make links between their mathematical understanding and their scientific knowledge. Eight-year-old Adam's comment illustrates this: "I learned how far away and how bright some stars are. All stars are heaps of light years away except for the sun. The sun is the closest star to the earth. It is a medium-sized star."

Conclusion

The children's responses throughout the activities suggest aspects of multidigit number sense that need consideration in teaching mathematically gifted elementary students. For example, some children were unaware of the existence of large numbers: "I never knew that there was such a thing as one hundred thousand" (Shaun, 8 years). Others, such as 6-year old Sandy, were amazed at the number of digits needed to represent large numbers symbolically (e.g., 1,000,000), in contrast to their verbal name (one million). The visual impact of constructing a million (meter cube) was evident in the comments of other children, such as Bryan (6 years) who recorded "I learnt that numbers could take up so much space."

The activities presented here fulfilled our aims of developing children's multidigit number sense to facilitate their understanding of the space travel resource material. At the same time, the children developed a fascination for large numbers, and derived enjoyment from conducting mathematical investigations. Additionally, there were opportunities for the children to develop their logico-mathematical intelligence and spatial intelligence (Gardner, 1983; Kruteskii, 1976). Most importantly, however, the children's reflection on their learning empowered their work as mathematicians. For example, Drew (8 years) commented, "I learned that if you think, you can figure out how to count large numbers [emphasis added]."

Although mathematically gifted children are characterized by the quality of their reasoning (Johnson, 1983), these children require appropriate and challenging learning experiences to facilitate their cognitive development (Henningsen & Stein, 1997; Hoeflinger, 1998). Enrichment that consists of "busy work" or irrelevant topics has limited academic value for gifted students. Although a student might be gifted, he or she still needs appropriate teacher support in dealing with challenging tasks that extend mathematical understanding. Enrichment programs play a key role in equipping these students with the foundational knowledge and skills to actively pursue their interests.

Manuscript submitted September, 1999. Revision accepted March, 2001.

(1) Clement, Rod. "Counting on Frank." Sydney: Angus & Robertson, 1990. Reprinted with permission from HarperCollins Publishers.

REFERENCES

Clement, R. (1990). Counting on Frank. Sydney: Angus & Robertson.

Gardner, H. (1983). Frames of mind: The theory of multiple intelligences. London: Heinemann.

Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524-549.

Hoeflinger, M. (1998). Developing mathematically promising students. Roeper Review, 20, 244-247.

Johnson, M. L. (1983). Identifying and teaching mathematically gifted elementary school students. Arithmetic Teacher, 30(5), 25-26.

Jones, G., Thornton, C., & Putt, I. (1994). A model for nurturing and assessing multidigit number sense among first grade children. Educational Studies in Mathematics, 27(2), 117-143.

Kruteskii, V. A. (1976). The psychology of mathematical abilities in school children. Chicago: University of Chicago Press.

Lupkowski-Shoplik, A. E., & Assouline, S. G. (1994). Evidence of extreme mathematical precocity: Case studies of talented youths. Roeper Review, 16, 144-151.

National Council of Teachers of Mathematics (1998). Principles and standards for school mathematics: Discussion draft. Reston, VA: National Association of Teachers of Mathematics.

Schwartz, D. (1985). How much is a million? New York, NY: Mulberry.

Dr. Carmel Diezmann lectures in mathematics education at Queensland University of Technology in Brisbane. She is a former elementary teacher with a long involvement in the education of gifted elementary children. Lyn English is a Professor of mathematics education at Queensland University of Technology in Brisbane. Her interests lie in the areas of mathematical reasoning, problem posing, and problem solving.