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558 May2011•teaching children mathematics www.nctm.org

Understanding

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www.nctm.org teaching children mathematics • May 2011 559

Developing an understanding of place value and the base-ten number system is considered a fundamental goal of the early primary grades (NCTM 2000, 2006). For years, teachers have anecdotally reported that students struggle with place-value concepts. Among the common errors cited are misreading such numbers as 26 and 62 by seeing them as identical in mean-ing, failing to recognize that numbers can be

composed of tens and ones, and incorrectly believing that large digits result in larger values regardless of the position of the digit (Kari and Anderson 2003; Sharma 1993). Results from a number of research studies support the notion that students lack an understanding of place value and have diffi culty distinguishing between the values of the tens and ones places (Hanich et al. 2001; Jordan and Hanich 2000; Kamii 1985,

By Linda L. Cooper and Ming C. Tomayko

Exploring ancient numbers helps students focus on the structure and properties of

our Hindu-Arabic system.

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1989; Kamii and Joseph 1988; Ross 1986, 1989; Ross and Sunflower 1995).

Kamii (1985) and Kamii and Joseph (1988) found this to be most prevalent in first grade, where none of their study’s students cor-rectly identified that the numeral 1 in 16 corresponds to a physical representation of ten ones; instead they indicated that the 1 simply corresponds to one unit. Hanich and her colleagues (2001) and Jordan and Han-ich (2000) reported similar findings among second-grade students. This misconception persisted through fourth grade, where only 50 percent of the students could associate the tens digit with the appropriate physical repre-sentation (Kamii 1985; Kamii and Joseph 1988; Ross 1986, 1989). Although students are expected to understand place value as early as second grade, these results indicate that true understanding does not occur until much later. An underdeveloped understanding of

place value can have far-reaching consequences. Notably for the elementary school student, it can inhibit and slow conceptual understanding of the basic arith-metic algorithms, whose repre-sentations rely on a place-value foundation.

During a summer enrichment program that we developed and taught, we set out to strengthen understanding of place value through a series of activities that put students in the unfamiliar set-ting of working with two ancient number systems (Egyptian and Mayan) whose rules differed from our own. The participants were three students from a subur-ban elementary school, who had just finished either second grade while working above-grade level or third grade working at grade level. The disposition of these students ranged from being inse-cure and unenthusiastic to self-assured and passionate about mathematics. Their parents had asked us to arrange a summer mathematics program that would engage the children. Aware of the

Students were given numbers and asked to draw the equivalent Egyptian symbol.

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47

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248

5309

Each child created an Egyptian mystery number and kept the solution a secret. After challenging one another, they later compared answers.

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To explore place value, students decorated and baked cookies with representations of Mayan number symbols.

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benefits of reinforcing place-value concepts, concepts that these second- and third-grade students, regardless of mathematics ability, were all still developing, we chose to develop activities around ancient number systems. We conjectured that students’ unfamiliarity with systems other than their own base-ten system would necessitate that they focus their attention on the underlying properties of these “new” systems. By learning how to read and write numbers in a system without place value and one whose use of place value differed from our own, students’ understand-ing of place value would be enriched, both globally and specifically to its role within our own Hindu-Arabic number system. Finally, we anticipated that students would respond enthusiastically to these explorations and would be more amenable to discussing place value in these new surroundings.

The activities we describe could be modi-fied for various audiences who would ben-efit from deepening their understanding of place-value concepts: as a supplement to the curriculum for primary-grades students, as center activities for more independent inter-mediate-grades students, and as an appeal-ing connection to discussions of ancient civilizations in social studies. In addition to enrichment activities, these have the poten-tial to challenge place-value misconceptions held by students whose understanding of our own number system is largely limited to rote procedures.

The egyptian number systemOur first effort to better understand the mean-ing and use of place value was to look at a number system that has no place value. The Egyptian number system, a base-ten system, uses the additive property, where the value of a number is determined by summing the values of the symbols that collectively represent that number. Although sources (Billstein, Libeskind, and Lott 2007; Leimbach and Leimbach 1990) differ as to whether order is important, we told our students that order of symbols does not matter. We first gave them Egyptian numbers (see table 1) and asked them to find the equiva-lent Hindu-Arabic number. Then, given Hindu-Arabic numbers, students were to write the numbers in the Egyptian system (see fig. 1).

A mystery-number activityAfter all three students practiced converting Egyptian numbers to Hindu-Arabic equiva-lents, we asked them to use Egyptian sym-bols to construct a number for their peers to convert. They attended to the task with great enthusiasm as they tried to stump one another. Each student created an Egyptian mystery number and kept a separate record of the solution (see fig. 2). They swapped papers and quickly got to work interpreting the Egyptian numbers. Once completed, the student who created the mystery number checked the class-mate’s solution against his or her own.

Writing numbersObserving that the students picked Egyptian numbers with many symbols for the mystery number activity, we segued into a discussion about a disadvantage of the Egyptian number system. We asked, “Can you think of a number that would not be easy to write using Egyptian symbols?”

At first, the children thought that numbers of larger values would be more difficult to write. To focus their attention on the issue of effi-ciency, we asked them to write the numbers 9 and 21 in both Hindu-Arabic and Egyptian and then compare them. Although writing

21 versus ∩∩ |

The ancient Egyptian number system used everyday symbols that were familiar to the culture.

Hindu-Arabic no.

Egyptian symbol

Explanation of symbol

1 a staff

10 an arch

100 a coiled rope

1000 a flower

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did not involve much difference, Riley noted that writing the number 9 in Hindu-Arabic is faster than writing it in Egyptian. We probed further by asking why it is faster. Riley responded, “There were fewer symbols to write.” He counted and found that writing 9 in Hindu-Arabic requires only one symbol but that writing the equivalent Egyptian number

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requires nine symbols. Students saw that the larger the value of the digits, the more cumber-some it is to represent the number in Egyptian.

Hindu-Arabic numbers containing the numer-als 8 and 9 are particularly inefficient to write in Egyptian. The students compared writing 248 in both systems and found that it took fourteen symbols to write the number in Egyptian yet only three symbols in their own Hindu-Arabic system (see fig. 3). For additional descriptions of Egyptian numbers (and the Mayan numbers to follow), refer to Billstein et al. 2007, Leim-bach and Leimbach 1990, and the Internet sites listed in the appendix.

The Mayan number systemThe Egyptian number system provided these three students with one lens to understand place value as they investigated how a number system works when there is no place value. For comparison purposes, we then turned our attention to the Mayan number system, which has the property of place value, but not in the way with which our students were familiar. Unlike our base-ten system, which uses ten digits, the Mayan system is a modified base-twenty system composed of only three symbols: a dot, a line, and a clamshell. Instead of a true base-twenty system with place-value positions of 1, 20, and 202, the Mayans used place-value positions 1, 20, and 360 because of their strong cultural interest in the lunar calendar. For our purposes, we focused only on the ones and twenties positions. Furthermore, whereas our place-value positions are listed horizontally in increasing order from right to left, the Mayan place-value positions are writ-ten vertically in increasing order from bottom to top, beginning with the ones position, fol-lowed by the twenties position.

At first we had our students concentrate only on mastering representations of the val-ues one through nineteen using the ones posi-tion (see table 2). In the ones position, each dot represents “one.” Dots are listed side by side in a row, with a row containing, at most, four dots. A line represents five “ones.” Lines are stacked one on top of another to represent multiples of five, with remaining dots—each with a value of one—placed above the lines.

Place-value positionsOur introduction of the second place-value position in the Mayan system, the twenties position, was the first time that these students

The three summer school students mastered Mayan numbers 1 to 19 using only the ones position.

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• = 11

• = 16

• = 11

• = 16

•• = 12

•• = 17

•• = 12

•• = 17

••• = 13

••• = 18

••• = 13

••• = 18

•••• = 14

•••• = 19

•••• = 14

•••• = 19

• = 15

• = 10

• = 15

Comparing the number of symbols in Hindu-Arabic with Egyptian showed the children how unwieldy some Egyptian representations are.

How many numerals are needed to write the number 248 in the Hindu-Arabic and Egyptian number systems?

Hindu-Arabic EgyptianNumber itself 248

Number of digits 3

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The Mayan representation of 48 illustrates how symbols in the twenties position are placed above symbols in the ones position.

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Twenties position (2 × 20 = 40)

Ones position (1 × 5 + 3 × 1 = 8)


had been exposed to place value other than in their own base-ten system. We asked, “Is it just coincidence that both our system and the Egyptian system are base-ten systems?” To clarify, we added that both systems have “spe-cial representations for tens and hundreds (ten tens). What is special about the number ten?”

Picking up on our use of the term base-ten, Casey proposed that our system is called a base-ten system because it has ten digits. We responded, “But why does our system have ten digits [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]? Why not only four digits [0, 1, 2, 3]?”

Finally, we asked what the Egyptians had in common with the people who developed the Hindu-Arabic number system that has to do with the number ten. Riley suddenly exclaimed, “Ten fingers.”

We followed up, “If counting on fingers led to a base-ten system, what might have motivated the Mayans to use a base-twenty system?”

With only a slight hesitation, Casey offered, “Ten fingers and ten toes.” Collectively, we sur-mised that it would be much easier to count to twenty using fingers and toes if you were not wearing shoes. (The Mayan grouping of five dots into a line may stem from having five digits on a hand or a foot.)

In the twenties position, each dot represents the value 20, and a line is equivalent to 5 × 20 = 100. Parallel to the ones position, there can be four dots at most, and lines are stacked verti-cally with dots placed above. Symbols in the twenties position are placed above symbols in the ones position, leaving a significant gap between positions. We initially chose to have the children use a frame to delineate the ones and twenties positions (see fig. 4).

Deciding how to write the number 20 led the group to discuss an important function of a special symbol within a place-value system, namely the zero symbol, represented by a clamshell in the Mayan system. Riley asked, “How do you know if one dot represents one or twenty?”

We acknowledged how important this ques-tion is and returned a question of our own: “How do we represent one ten and no ones in our own system?”

Casey answered, “That’s just ten.” Jordan added, “You need a zero.” The Mayans used a clamshell to represent

zero. Jordan, Casey, and Riley realized that without a way to represent zero ones, it would be impossible to differentiate whether a dot above a line were symbols in the ones position, equivalent to the number 6, or in the twenties position, equivalent to the number 120:

Our next activity was to have the students read and write Mayan numbers that involve both ones and twenties place-value positions. Given a Mayan number represented by lines, dots, and possibly a clamshell, the children separately found the values in the ones posi-tion and then in the twenties position, finally summing to find the value of the number as a whole. It proved to be a far more challenging task for them to start with the Hindu-Arabic representation and translate into the equiva-lent Mayan representation: They had to deter-mine how many groups of twenty were in the number, represent the number of twenties using the system of dots and lines, subtract the determined multiples of twenty from the original number, and finally represent the remainder with dots and lines in the ones posi-tion. Similar to our activities with the Egyptian number system, the students wrote mystery numbers and challenged their classmates to find the Hindu-Arabic equivalent (see fig. 5).

Ordering Mayan number treatsBeing able to make comparisons of magnitude is a vital skill when working within a number system. Our students ordered Mayan numbers that served as cookie decorations. The purpose of this activity was to have students discover how their understanding of Mayan place value

Just as they had done with Egyptian numbers, the children wrote Mayan mystery numbers to challenge one another.

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The teachers challenged the children to arrange cookies in ascending order and then look for patterns in their sequence.

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Comparing number systemsTeachers can use such questions as the following to prompt discussions about number systems with and without place value:

• Whatspecial number is needed to hold a place-value position in a system with place value? Think about how you would write such a number as three hundred fi ve. Do you need a special number in a system without place value?

• Whichtype of system is more effi cient when writing numbers? In other words, in general, does it take more symbols or fewer symbols to write a number in a system with place value or one without place value? Think about how to write such numbers as forty-three, seventy-eight, and ninety-nine in Egyptian, Mayan, and Hindu-Arabic. Which one of these three systems appears to be the most effi cient? Which is the least effi cient?

(presence or lack of symbols in the twenties position and presence or lack of lines within a place-value position) allowed them to quickly make comparisons of magnitude. We provided cookie dough, miniature chocolate chips, and narrow chocolate strips and asked the chil-dren to decorate cookies with Mayan numeric symbols on them. The chocolate chips rep-resented dots, and the chocolate strips represented lines. We did not provide for a clamshell. In addition to the students’ cook-ies, we made additional cookies decorated with numbers of larger values. After baking the dough, we mixed up the cookies, placed them on a tray (see fi g. 6), and challenged students with the task of working together to arrange the cookies in ascending order. Initially, the process was slow; students chose to determine the equivalent Hindu-Arabic value of each number to make comparisons. Some cookies could be rotated 180 degrees to result in a dif-ferent but valid Mayan number. After students completed the task, we had them look for pat-terns in their sequence. They noticed that the fi rst numbers in their sequence, those of value less than twenty, did not have symbols in the top portion of the cookie, whereas those cook-ies with numbers of value twenty or greater did. To see if this realization made a difference in the ease of ordering numbers, we mixed up the cookies and asked students to reorder them. This time, students were able to com-plete the task more quickly (see fi g. 7). They had learned how to fi rst order by the presence or lack of value in the twenties position, much as students recognize that a Hindu-Arabic number containing a digit in the hundreds position must be greater than a number with only two digits.

globetrottingOur investigation of number systems pre-sented a novel opportunity to make connec-tions between mathematics and social studies. As we introduced each number system, we encouraged students to connect the systems to their geographical and historical origins. We made the important point that different cultures create their own ways of representing numbers and mathematical ideas. In addition to our classroom globe, students had world map outlines on which they could each mark

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geographical areas of interest (see fi g. 8). They located Egypt and associated it with the great pyramids, the Nile River, and the ancient pharaohs. We turned the globe and guided students to the Mayan civilization’s location in present-day Mexico and Guatemala. They found the Mayans’ similar use of pyramids par-ticularly intriguing because they were unaware that pyramids had also been constructed in the Western Hemisphere. The online appendixprovides Internet links and directions to locate ancient pyramid ruins using Google Earth as well as other Web sites that provide historical background about these civilizations.

The Hindu-arabic systemWe next focused our discussion on the spread of the Hindu-Arabic number system. We asked our students to think about the name Hindu-Arabic, which describes where the number system developed. Specifically, we asked which present-day country is associated with the word Hindu. Students readily associated Hindu with India. After being guided to the general geographical area, they fi rst located India on the globe and subsequently identifi ed it on their outline maps. We explained that the second part of the name, Arabic, describes the path that the number system took as it spread (and evolved). We asked students to fi nd the Arabian Peninsula and gave them the hint that Saudi Arabia is the largest present-day coun-try on that peninsula. Whereas our students had some familiarity with India, including its shape, we found that we needed to assist them in fi nding Saudi Arabia and the Arabian Peninsula. Students labeled Hindu on India and Arabic on the Arabian Peninsula and drew a land path to connect them (see fi g. 8). Show-ing them where several familiar European countries are located, we then asked how the

Noticing that numbers lower than twenty had no symbols in the top portion of the cookie helped students complete the same task more quickly the second time.

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Using outline maps of the world, Jordan, Riley, and Casey learned that different cultures create their own ways of representing mathematical concepts.

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number system could have spread to there. Students offered several paths, some by land and others traversing the Mediterranean Sea. Finally, we asked students how they thought the use of the Hindu-Arabic number system spread from Europe to the United States. With-out hesitation, Jordan responded “Columbus.”

Yes, we acknowledged, Columbus and the many other explorers and settlers from Spain, England, Portugal, and so on would have brought their Hindu-Arabic number system with them. The journey of our number system is just one example of how cultures spread.

Souvenir charmsFor our fi nal activity, we hoped that student-created souvenir charms would serve as reminders of our explorations into ancient number systems. Each student received pieces of number 9 polystyrene plastic (7 × 7 cm) and a variety of colors of permanent mark-ers. Students were to write the names of each ancient number system and provide examples of numbers from those systems. Some students


included the Hindu-Arabic equivalents. We punched holes in the charms of those students who wanted the option of stringing them, and then we baked the charms in a toaster oven at 350 degrees Fahrenheit for thirty seconds. (Heating polystyrene in a toaster should be done in a well-ventilated area and only with adult supervision.)

extending the exploration The Egyptian and Mayan systems are both characterized by the repetition of symbols, whether they are staffs and arches or dots and lines. In each case, the process of combin-ing symbols is reminiscent of the regrouping step that children use when physically adding base-ten blocks. Adding the Egyptian numbers 45 + 57 involves combining twelve staffs to

form one group of ten staffs, which is replaced with an arch and two leftover staffs.

∩∩∩∩||||| + ∩∩∩∩∩|||||||

However, a clear distinction exists between the Egyptian and Hindu-Arabic systems. Whereas the use of base-ten blocks physically models and parallels the process of adding as notated by our traditional algorithm in the Hindu-Arabic system, a lack of place value precludes a corresponding shorthand notation in the Egyptian system. Adding in Egyptian is simply regrouping symbols.

Adding numbers in the Mayan system involves a two-tier regrouping process. Start-ing within the ones position, dots are com-bined, and groups of five dots are replaced by a line. A sum that results in more than five lines in the ones position results in each group of five lines from the ones position being replaced with a dot in the twenties position. Once again, within the twenties position, groups of five dots are replaced with a line.

Concluding remarksOur exploration of Egyptian and Mayan number systems resulted in students mak-ing several connections between the ancient systems and our own. Counting on ten fingers and ten toes led to number systems based on ten (Egyptian and Hindu-Arabic) or twenty (Mayan). Before this experience, students took for granted that numbers are always written horizontally, with the smallest place-value position to the extreme right. Writing Mayan numbers was at first a game as stu-dents learned the use of symbols and vertical place-value positioning in that system. As they became more familiar with the Mayan system, they began to make connections and realized that our way of positioning place values is not unique. If they had been Mayan, they would have written their numbers vertically. The use of the Mayan clamshell drove home the signif-icance of our own zero: Students realized that without it, they were unable to determine the place-value position of symbols, resulting in a representation of a number with ambiguous value. Students realized that both a numeral and its position affect a number’s value. Being

TOP: This child included Hindu-Arabic equivalents when drawing numbers from other systems on the charms. Before baking the charms, the teachers punched holes in them for the students who wanted to string their charms together.

BOTTOM: Students marveled as their charms shrank in size after thirty seconds in a hot oven.

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able to see that the dot in the Mayan twenties position represents one twenty, and not just one, strengthened their understanding that the one in our tens place represents one ten, and not just one. At first the Egyptian system seemed easier to work with, but students later appreciated that a place-value system such as our own requires fewer symbols to represent a value. (See sidebar on p. 564 for key questions to ask when comparing number systems with and without place value.)

This investigation truly engaged our stu-dents; some wanted to continue the lesson and learn how to add in the ancient systems. As Jordan enthusiastically exclaimed, “This is a neat type of math!”

REFERENCESBillstein, Rick, Shlomo Libeskind, and Johnny W.

Lott. A Problem-Solving Approach to Mathe-matics for Elementary School Teachers. Boston: Pearson/Addison Wesley, 2007.

Hanich, Laurie B., Nancy C. Jordan, David Kaplan, and Jeanine Dick. “Performance across Different Levels of Mathematical Cognition in Children with Learning Difficulties.” Journal of Educational Psychology 93, no. 3 (September 2001): 615–26.

Jordan, Nancy C., and Laurie B. Hanich. “Math-ematical Thinking in Second-Grade Children with Different Forms of LD.” Journal of Learn-ing Disabilities 33, no. 6 (November-December 2000): 567–78.

Kamii, Constance. Young Children Reinvent Arith-metic: Implications of Piaget’s Theory. New York: Teachers College Press, 1985.

———. Young Children Continue to Reinvent Arithmetic, 2nd grade: Implications of Piaget’s Theory. New York: Teachers College Press, 1989.

Kamii, Constance, and Joseph, Linda. “Teaching Place Value and Double-Column Addition.” Arithmetic Teacher 35, no. 6 (1988): 48–52.

Kari, Amy R., and Catherine B. Anderson. “Oppor-tunities to Develop Place Value through Student Dialogue.” Teaching Children Mathematics 10, no. 2 (2003): 78–82.

Leimbach, Judy, and Kathy Leimbach. Can You Count in Greek? Exploring Ancient Number Sys-tems. San Luis Obispo, CA: Dandy Lion Publica-tions, 1990.

National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000.

———. Curriculum Focal Points for Prekindergar-ten through Grade 8 Mathematics: A Quest for Coherence. Reston, VA: NCTM, 2006.

Ross, Sharon Hill. The Development of Children’s Place-Value Numeration Concepts in Grades Two through Five. Paper presented at the 67th Annual Meeting of the American Educa-tional Research Association, San Francisco, CA, April 1986.

———. Parts, Wholes, and Place Value: A Devel-opmental View. Arithmetic Teacher 36, no. 6 (1989): 47–51.

Ross, Sharon Hill, and Elisa Sunflower. Place-Value: Problem Solving and Written Assessment Using Digit-Correspondence Tasks. Paper presented at the 17th Annual Meeting of the North Ameri-can Chapter of the International Group for the Psychology of Mathematics Education, Colum-bus, OH, October 1995.

Sharma, M. “Place-Value Concept: How Children Learn It and How to Teach It.” Math Notebook 10, nos. 1 and 2 (January 1992–February 1993): 1–23.

Linda L. Cooper, [email protected], and Ming C. Tomayko, [email protected], are colleagues at Towson University in Towson, Maryland, where they teach content and methods courses in mathematics education to preservice and in-service teachers of grades K–12.

An appendix accompanies the online version of this article at www.nctm.org/tcm.

The NCTM 2011 Board of Directors Election will be held online this fall. No paper ballots will be mailed; so make sure to keep your membership proole information up-to-date to ensure you receive e-mail notiications and election information. Log on today—www.nctm.org/proile or call 800-235-7566.

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➺ appendix to “Understanding place value”

May 2011 • teaching children mathematics www.nctm.org

Egyptian and Mayan sites

• For an outline map of the world: http://www.eduplace.com/ss/maps/pdf/world_cont.pdf

• For a bird’s-eye view of ancient ruins: www.maps.google.com(Zoom out to obtain geographical reference; zoom in to see sites. Click on markers for street-view photographic images of sites.)

• Enter address “Cairo-El Rayoum, Giza, Egypt” to view the Great Pyramids.

• Enter address “Templo de Kukulcan, Chichen Itza, Mexico” to view Mayan ruins.

Egyptian sites

• Number translator: http://www.psinvention.com/zoetic/tr_egypt.htm

• Background information for numbers, brief quiz using Egyptian mathematics, pyramids, videos, and more: http://www.eyelid.co.uk/numbers.htm

Mayan sites

• Background information for Mayan numbers and number translator: http://gwydir.demon.co.uk/jo/numbers/maya/index.htm#count

• Comparison of the Mayan number system to the Hindu-Arabic and Roman number systems, including operation of addition: http://mathforum.org/k12/mayan.math/index.html

• Background information for Mayan numbers and addition quiz: http://www.niti.org/mayan/lesson.htm

Informative and Useful Internet Links

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