Visualization and Explicit Number Naming as a Foundation

for Children's Early Work in Mathematics

PbyD Paper for WGA1, Aug. 2000

 

Joan A. Cotter, Ph.D.

International studies, such as the TIMSS studies, show that Asian students score higher in mathematics than their American counterparts. In the U.S. half the children in fourth grade are still learning place value concepts (Ross, 1989, Miura & Okamoto 1989); whereas, Asian children develop this concept years earlier. There are several Asian practices that could greatly benefit U.S. and many other children. These are minimizing counting, visualizing quantities, explicit number naming, and carefully choosing manipulatives

 

The Counting Model

The counting model, prevalent in many western countries, requires mastery of one-to-one correspondence, which generally does not occur until after age 6. Young children upon hearing the words "how many" start the counting ritual. The child’s understanding of magnitude is hazy, merely a vague recollection of the length of the list of counting words. Some children even decide "greater than " problems based on counting with the number spoken first as the lesser of the two. A considerable delay occurs in the child’s recognizing cardinality as the last number counted.

Children taught counting to find sums frequently find it difficult to advance to more efficient strategies. Children who count to add and subtract develop a unitary concept of number. That is, they think of 14 as 14 ones, not as 1 ten and 4 ones. Such thinking interferes with understanding carrying and borrowing in multidigit numbers.

 

The Visualization Model

The visualization model involves developing mental patterns for quantities. Wynn (1992) discovered that five-month-old babies could add and subtract up to three objects. The researcher showed the baby several teddy bears and then raised a screen where the number could be manipulated undetected by the baby. The researcher then showed the baby, for example, another teddy bear and placed it behind the screen, and then lowered the screen. Gaze time, the length of time the baby looked, was significantly greater for incorrect numbers of teddy bears up to three, than for the expected number. Obviously, the babies were not counting.

Virtually no one can visualize more than five objects without grouping. Try to see mentally 8 apples in a line without any grouping&emdash;virtually impossible. Now try to see 5 of those apples as red and 3 as green; the vast majority of people can form the mental image. The Romans, who originally wrote 4 as "IIII," introduced a "V" for 5 to avoid counting. Western music is written with five-line staffs. When ten lines are needed, as in piano scores, the two groups of fives are separated to make the music readable.

Fingers are a natural fives grouping. Teach the child the names for quantities 1-5 by asking the child to raise the appropriate number of fingers with the left hand. (Numbers are read from left to right.) Show quantities 6-10 with 5 on the left hand and remainder on the right hand. Do this without any counting, only naming the quantities.

A simple manipulative is "tally sticks." Tally marks are used in Asia, Europe, Australia, North America, and South America. Tally sticks are simply Popsicle or wooden craft sticks and measure 1 cm by 11 cm. Show the child how to represent quantities 1-4 by laying the sticks vertically about 2.5 cm apart. Demonstrate 5 by laying the fifth tally stick horizontally across the center of the first four; avoid diagonals. Continue with 6-10 the same way; however, start a new row for each ten.

 

Explicit Number Naming

English has many inconsistencies in number naming. Ten has three names, ten, -teen, and -ty. Eleven and twelve seem to make no sense and for the numbers 13-19, the order is reversed with the ones stated before the tens. All Indo-European languages have irregularities in number naming. German reverses the order of tens and ones through 99.

On the other hand, Asian languages use explicit number naming. Counting beyond ten is ten-1, ten-2, ten-3 for 11-13; 2-ten 1, 2-ten 2, and 2-ten 3 for 21-23; and 9-ten 9 for 99. It takes 28 words to count to 100 in English, but only 11 words in an Asian language.

A natural experiment in number naming occurs in Korea, where they use two number systems. For informal, or everyday, speech the number words are irregular, but in school the formal number system is explicit and completely regular. No number words are the same in the two systems. At age 4, Korean children on the average trailed U.S. children in their ability to count; Korean children counted to 4 in both systems and U.S. children counted to 20 (Song and Ginsburg, 1988). Between ages 5 and 6, the Korean children’s counting ability jumped from 28 to 90 in the formal system and from 20 to 60 in the informal system. Between the same ages of 5 and 6, the U.S. children’s counting ability increased only a modest amount from 45 to 70, which was the same rate as between ages 4 to 5, indicating rote memorization.

Miura and Okamoto (1989) discussed the possibility that the Asian language system of explicit naming is one of the factors associated with the high mathematics achievement of Asian-American students. Data from the California Assessment Program (1980, 1982) as cited in Miura and Okamoto showed that Asian-American students scored higher in mathematics than other groups. When data from the 1979-80 year is grouped by language spoken, greater variations were seen. Asian-American third graders who spoke only English scored in the 54th percentile, while students who were also fluent in Chinese or Japanese scored in the 99th and 97th percentiles, respectively (Sells, 1982). This compares with bilingual Spanish-speaking third graders who scored in the 16th percentile. Young children speaking any Indo-European language can benefit from using explicit number naming.

 

Manipulatives

U.S. primary classrooms generally have an overabundance of manipulatives, many of which are counters, since they are not grouped in fives. Japanese primary classrooms have very few manipulatives, all of which the children must be able to visualize.

The AL Abacus. This double-sided abacus represents quantities in fives and tens. On side 1 each bead has a value of 1. In Figure 1, 7 is entered and in Figure 2, 76 is entered.

 

On side 2 of the abacus, beads have a value according to their position. Two wires are used to show each denomination. See Figure 3. Enter quantities by moving beads up. Trade 10 ones for 1 ten by entering 1 ten-bead while removing 10 one-beads.

Place Value Cards. Place-value cards help children construct and record multidigit numbers. See Figure 4. The 8 in 813 is 8 hundred because two zeroes (or other digits) follow it. These cards encourage the children to read numbers in the normal left to right order, rather than backwards as is done with the prevalent column approach of starting at the right and saying, "ones, tens, hundreds."

 

Research Study

Research was conducted in an experimental first grade class in which the children used visualization, explicit number naming, the AL abacus, and the place-value cards (Cotter, 1996). The control class learned in the traditional workbook method.

Some significant findings comparing the experimental class to the control class are the following: (a) Three times as often, the experimental class preferred to represent numbers 11, 13, 28, 30, and 42 with tens and ones instead of a collection of ones. (b) Only 13% of the control class, but 63% of the experimental class correctly explained the meaning of the 2 in 26 after the 26 cubes were grouped in 6 containers with 2 left over. (c) In the control class 47% knew the value of 10 + 3 and 33% knew 6 + 10, while 94% and 88%, respectively, of the experimental class knew. (d) In the control class 33% subtracted 14 from 48 by removing 1 ten and 4 ones rather than 14 ones; 81% of the experimental class did so. (e) When asked to circle the tens place in the number 3924, 7% of the control class and 44% of the experimental did so correctly. (f) None of the control class mentally computed 85 - 70, but 31% of the experimental class did. (g) For the sum of 38 + 24, 40% of the control class incorrectly wrote 512, while none in the experimental class did.

Notable comparisons with the experimental class and the work of other researchers showed that: (a) All the children in the made at least one "tens and ones" representation of 11, 13, 28, 30, and 42, while only 50% of the U.S. children did so in the study by Miura & Okamoto (1989). (b) 63% made all five "tens and ones" representations, while only 2% of the U.S. children did so in the study by Miura & Okamoto. (c) 93% of the children explained the meaning of the digits in 26 while only 50% of the third graders in Ross’s (1989) did so. (d) 94% of the children knew the sum of 10 + 3 while 67% of beginning second graders in Steinberg’s (1985) knew. (e) 88% knew 6 + 10 compared to 72% of the second graders in Reys et al. (1993) study. (f) 44% of the children circled the tens place in 3924 while data from the 1986 NAEP (Kouba et al., 1988) found 65% of third graders circled the tens place in a four-digit number. (h) 56% mentally computed 64 + 20, which compared to 52% of nine-year-olds on the 1986 NAEP study. (i) 69% mentally computed 80 - 30 while 9% of the second graders in Reys et al. study did so.

 

References

  • Cotter, J. A. (1996). Constructing a Multidigit Concept of Numbers: A Teaching Experiment in the First Grade. (Doctoral dissertation, University of Minnesota, 1996). Dissertation Abstracts International #9626354.

    Kouba, V. L., Brown, C. A., Carpenter, T. P., Lindquist, M. M., Silver, E. A., & Swafford, J. O. (1988). Results of the Fourth NAEP Assessment of Mathematics: Number, Operations, and Word Problems. Arithmetic Teacher, 35(8), 14-19.

    Miura, I. T., & Okamoto, Y. (1989). Comparisons of U.S. and Japanese First Graders' Cognitive Representation of Number and Understanding of Place Value. Journal of Educational Psychology, 81(1), 109-114.

    Ross, S. H. (1989). Parts, Wholes, and Place Value: A Developmental View. Arithmetic Teacher, 36(6), 47-51.

    Sells, L. W. (1982). Leverage for Equal Opportunity Through Mastery of Mathematics. In S. M. Humphreys (Ed.), Women and Minorities in Science (pp. 6-25). Boulder, CO: Westview Press.

    Song, M., & Ginsburg, H. (1988). The Effect of the Korean Number System on Young Children's Counting: A Natural Experiment in Numerical Bilingualism. International Journal of Psychology, 23, 319-332.