<<

The Chinese Numeration and Place Value

chi, ni, san . . . Otu, abuo,ato. . . Eins, zwei, drei . . . Have you heard these words before? What might

they mean? In case you are not familiar with these words, they mean “One, two, three . . .” in Japan- I ese, Igbo, and German, respectively. Two of the most fundamental topics in mathe- the unique needs and purposes of those who used matics that children must learn are counting and them (Cathcart et al. 2000). These needs ranged the numeration system. Central to the and from tallying the number of animals kept to devel- Operations Standard of NCTM’s Principles and oping calendars. Many cultures created culturally Standards is the development of number sense, specific numeration that included their which includes the ability to understand the base- own , rules, and values. One of these is the ten number system (NCTM 2000). Reys (1998) Chinese numeration system. explains that place value is the foundation of our The Chinese numeration system is a numeration system. We use place value to read, (base-ten) system, unlike other systems such as the represent, and operate on and also to iden- Babylonian ( or base-sixty) or the tify patterns, build number sense, and understand Mayan ( or base-twenty). In the different numeration systems. This article presents decimal system, counting is based on ten the Chinese numeration system as an additional numerals, symbols are used to represent the Frederick L. Uy representation of numbers and place value for numerals, and successive place values— teachers to consider in their instruction. Sample such as 10; 100; 1,000; and 10,000—are learning activities using the Chinese numeration powers of ten (Baumgart et al. 1989). Figure 1 system for elementary students are provided. shows the characters used in the Chinese numera- tion system. The Chinese Numeration

System Explained Frederick Uy, [email protected], is an assistant professor of mathe- matics education at State University in Los Angeles. His Virtually every civilization and culture developed research interests include ethnomathematics, multicultural , the concept of numbers and the formulation of a and geometry. counting process (Eves 1990). Cultural differences led to the creation of numeration systems suited to

JANUARY 2003 243 Copyright © 2003 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM. What is unique about this numeration system? The written form indicates the number of units in each base, together with that base. For example, the number 72 is actually written as seven ten(s) and two ( ). In contrast, in the Hindu- system, the written numerals indicate digits that are positioned accordingly, and each digit is a repre- sentation of a multiple of some power of the chosen base (Eves 1990). Consider the number 564. The “5” has a value of 5(102), or 500; “6” has a value of 6(101), or 60; and “4” has a value of 4(100), or 4. In the Hindu-Arabic system, the base (10) is implied. Written in Chinese characters, 564 is (5; 100; 6; 10; and 4), which liter- ally means five hundred(s), six ten(s), and four. In the Chinese numeration system, correspond- ing characters exist for 0Ð9 and for the multiples of 10, that is, 10; 100; 1,000; and so on. A number can easily be rewritten from Hindu-Arabic to Chinese by writing it in its expanded form, which gives the actual value of each digit. For primary students, it is suggested that a number be expanded as a sum of its parts (Cathcart et al. 2000). An example is 745 = 700 + 40 + 5, or 7 hundreds + 4 tens + 5. For intermediate students, the can be used: 745 = (7 × 100) + (4 × 10) + 5 or (7 × 102) + (4 × 101) + (5 × 100). Consider the number 15 and its expanded form,

10 + 5. Using the table in figure 1, 10 is represented Uy;Vanessa by Photograph all rights reserved as and 5 as . Therefore, 15 written in Chinese is . These concepts also can be will appear as . Another example is 79. In extended to numbers in the thousands and ten thou- expanded form, 79 = 70 + 9. In Chinese, 70 is writ- sands. If written properly, 2,745 in Chinese will ten as and 9 as , and the combination of the appear as (two thousand[s], seven two is . The principle is the same for greater hundred[s], four ten[s], and five) and 39,681 as numbers: 193 = 100 + 90 + 3. Putting the corre- (three ten-thousand[s], nine sponding characters together for these numbers, 193 thousand[s], six hundred[s], eight ten[s], and one).

The Chinese numeration system

0zero 10 ten

FIGURE 1 1 one 100 one hundred 2two 1,000 one thousand 3 three 10,000 ten thousand 4four 100,000 one hundred thousand 5 five 1,000,000 one million 6 six 7seven 8 eight 9 nine

244 TEACHING CHILDREN MATHEMATICS A natural question that will arise is, what if there is a zero digit? If the zero digit is at the end of the Activity 1: Complete the table below. number, there is no issue. 5,300 will be written as 5,000 + 300; in Chinese, this is (five Number Expanded Form Chinese System thousand[s], three hundred[s]). The more interest- (a) 398 300 + 90 + 8 ing scenario is when the zero digit lies in the mid- FIGURE 2 (b) 751 700 + 50 + 1 dle of the number. Consider 6,028 and its expanded (c) 576 form of 6,000 + 20 + 8. In a case such as this, it is (d) 307 300 + 0 + 7 recommended that zero be considered a part of the (e) 4,000 + 500 + 80 + 2 expanded . Therefore, 6,028 is better writ- ten as 6,000 + 0 + 20 + 8. Using the Chinese char- () 7,500 acters correspondingly, 6,028 now looks like (g) 80,427 . Try writing 90,999 in Chinese. You (h) 38,074 should get . For sample class- room activities, see figures 2Ð4. Benefits of the Lesson The Field Test Several benefits of this instructional approach were The activity in figure 2 was tried in a fourth-grade noted. First, the activities reinforced the students’ class in Rio Vista Elementary School in Pico knowledge of place value. According to the Number Rivera, California. The students were Hispanic, and and Operations Standard in Principles and Stan- English was not the primary at their dards for School Mathematics, instructional pro- homes. This lesson was chosen to serve as an exten- grams should enable students to understand num- sion of place value and to expose the students to a bers, ways of representing numbers, relationships different way to name numbers and express place among numbers, and number systems (NCTM value than they would find in either Spanish or Eng- 2000). These activities were aligned with the Stan- lish. Before doing the activity in figure 2, the stu- dard. Furthermore, students first wrote the given dents were given base-ten blocks to use with place- value activities. This served as a review of base-ten Activity 2: Use Chinese characters to write your answers. concepts, expanded form, and groupings that repre- sent numbers and acted as a bridge between con- crete and symbolic representations of numbers. The Questions Answers author and the teacher collaborated in introducing (a) What is the value of “8” in and demonstrating the Chinese numeration system FIGURE 3 4,834? to the students. After the lesson was conducted, stu- dents then were asked to complete Activity 1 and write about their understanding of place value. Two samples of student work are shown in fig- (b) What year were you born? ure 5. The first sample indicates that the student could translate from Hindu-Arabic to Chinese read- ily and had a procedural knowledge of the value of a digit, its place value, and its symbolic representa- tion. In the second sample, the student wrote the (c) What year did Arizona become Chinese representation for each digit without writ- a state? ing the place value, or simply left them blank. When asked why they did this, students who made this mistake said that they translated exactly the digits from Hindu-Arabic to Chinese and did not (d) What is the weight in ounces consider the place values of the digits. This served of an object weighing 1,032 as a flag for the teacher to inquire into the under- pounds? standing of these students. After doing the activity, some students still seemed to experience difficulty with place value and expanded forms of numbers. (e) How many times does your The teacher then attempted to clear up the confu- heart beat in three minutes? sion by modeling the place values, using concrete representations such as base-ten blocks while writ- ing the expanded for each number.

JANUARY 2003 245 Activity 3: Answer the following questions.

1. Your classmate wrote 4,732 as . Is this correct or incorrect? Explain. FIGURE 4

2. Write 378 in Chinese and how you will explain to a friend what the characters mean.

3. How do you think “ten million” is written in Chinese? Try it.

numbers in the expanded form before using Chinese conceptual knowledge of place value. Through the characters, demonstrating their understanding of the use of Chinese numeration representations, stu- relationships between the base-ten place values. The dents reviewed the base-ten place values in other students also reviewed the relationship between the contexts. Introducing a different cultural perspec- value of each digit and the total value of the numbers tive toward a mathematical concept shows students as recommended by Heddens and Speer (1997). that there are other ways to learn mathematics. The When asked how to write numbers in Chinese, stu- sample activities presented use a multicultural dents said, “First you write the for that num- approach. You are invited to try them and offer ber and then write the symbol for its value.” feedback about students’ mathematics learning A second benefit of the activity was that it cre- through them. ated a foundation for students’ algebraic thinking. Some other suggested activities include examin- The Chinese numeration system shows the number ing the origin and history of the Chinese numera- of units in each base, along with that particular tion system, researching other numeration systems, base. This thinking involves the concepts of coeffi- comparing numeration systems, and reading num- cients and “unknowns” (variables); that is, the bers in different . Another suggestion is number of units in each base represents the coeffi- to continue to have students write their reactions to cient, and the base is the variable. or reflections about the lesson. The students’ reac- Finally, this type of activity emphasizes the fact tions and reflections will give teachers insights into that mathematics permeates all cultures and soci- students’ understanding of and appreciation for eties. Principles and Standards for School Mathe- base-ten and other numeration systems. During the matics recognizes that mathematics is part of cul- study, one student noted that she finally understood tural heritage. It states further that mathematics is place value after being introduced to the Chinese considered to be one of the greatest cultural and numeration system. intellectual achievements of humankind, and an Students in today’s mathematics classes have appreciation and understanding of that achieve- diverse backgrounds, and we teachers should ment must be developed (NCTM 2000). employ techniques and methods that will accom- An earlier study conducted by the author con- modate their needs. Maintaining meaningful cluded that when a lesson is presented through a contexts for our lessons is essential. The use of cultural or historical connection, students see more the Chinese numeration system to enrich stu- of the value and use of mathematics in the world dents’ understanding of place value is only one outside the classroom and gain a better apprecia- example of the many cultural numeration sys- tion for the mathematics that arises (Uy 1996). A tems that you can investigate and share with multicultural approach to teaching mathematics your students. can enable students to take pride in the accom- plishments of their own people (Zaslavsky 1991). References Baumgart, John, Duane Deal, Arthur Hallerberg, and Bruce Vogeli, eds. Historical Topics for the Mathematics Class- Conclusion room. Reston, Va.: NCTM, 1989. The Chinese numeration system is one of many Cathcart, . George, Nadine Bezuk, Yvonne Pothier, and numeration systems that can reinforce a student’s James Vance. Learning Mathematics in Elementary and

246 TEACHING CHILDREN MATHEMATICS Sample student work FIGURE 5

Middle Schools. Upper Saddle River, N.J.: Merrill, 2000. Reys, Robert, Mary Lindquist, Nancy Smith, and Marilyn Eves, Howard. An Introduction to the Suydam. Helping Children Learn Mathematics. Needham with Cultural Connections. Orlando, Fla.: Saunders Col- Heights, Mass.: Allyn and Bacon, 1998. lege Publishing, 1990. Uy, Frederick. “Geometry in the Middle Grades: A Multicul- Heddens, James, and William Speer. Today’s Mathematics. tural Approach.” Unpublished doctoral dissertation. Upper Saddle River, N.J.: Merrill, 1997. Columbia University, May 1996. National Council of Teachers Mathematics (NCTM). Princi- Zaslavsky, Claudia. “Multicultural Mathematics Education ples and Standards for School Mathematics. Reston, Va.: for the Middle Grades.” Teacher 38 (February NCTM, 2000. 1991): 8Ð13. ▲

JANUARY 2003 247