Numeration Systems�A White Elephant

David Rappaport

Northeastern Illinois UniversityChicago, Illinois

When the new mathematics programs were introduced in elementaryschools more than a decade ago, the topic that made the greatest impactupon teachers, parents and children was numeration systems. Whenpeople at that time asked what is the new math all about, they wereusually told about the various bases. Here was visual evidence that the"new math" was really new. Many teachers were confused by thenumeration systems and reacted with amazement when childrendemonstrated their skill with base eight or base twelve arithmeticaloperations at professional meetings. Parents realized, some for the firsttime, that they were unable to help their children with their mathematicshomework.

Publishers were advertising their materials with such blurbs as "I + 1= 10." Authors were writing many articles about the differentnumeration systems.’ The large number of articles gave prestige to thistopic. It seemed that everyone was getting on this new bandwagon.During these early years every meeting of mathematics teachers was sureto have some talks on the numeration systems.Not everyone was happy with this new topic. Parents were asking why

their children were learning something that had no immediate practicalapplication. Of course, they were told that the rapid computers werebased on the binary system, but this seemed to be irrelevant for children,very few of whom would ever become computer operators. Childrenwere confused and could receive no help from their parents. Collegestudents in mathematics courses for teachers and in methods coursesoften rebelled at the requirement to learn the arithmetic skills in the dif-ferent bases. They often asked what.is wrong with base ten that they hadto learn other bases. If they had so much difficulty with the bases, thensurely the children would experience greater difficulty. Many elementaryschool teachers were very unhappy with this new topic that was forcedupon them by administrators and curriculum makers.That teachers were confused is illustrated by the following experience.

Several years ago I was asked to conduct a workshop for fourth, fifth

1. ArmandJ.Galfo, "When Does2 + 2 = 10?" School Science and Mathematics, (November 1963), pp.653-657.E.W. Hamilton, "Number System, Fad or Foundation?" The Arithmetic Teacher, 8 (May 1961), pp. 242-245.Charles Hudson, "Some Remarks on Teaching Different Bases," School Science and Mathematics, (November

1963), pp. 649-652.Ann C. Peters, "The Number System and the Teacher," The Arithmetic Teacher, 4 (October 1957), pp. 155-160.Emma L. Stringfellow, "Number Systems," School Science and Mathematics, (October 1959), pp. 557-560.

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28 School Science and Mathematics

and sixth-grade teachers of a suburban school system near Chicago. Thestandardized tests showed that the children in this school system rankedvery high on math concepts but were below average on arithmeticalskills. The principal suggested that, before the workshop began, I meetwith the math coordinators for these grades. The fourth-grade co-ordinator complained that she spent so much time getting children tolearn the bases that she had little time left in which to teach the othertopics. When I asked her why she taught the bases, she appeared to bedumbfounded. It was in the textbook so she was sure that it had to betaught. When the same question was asked of the fifth and sixth-gradecoordinators, they answered that they spent very little time with the basesbecause there were few pages in their textbooks devoted to this topic. Thefourth-grade teacher could have skipped this topic without any harmfuleffect upon the children for work in the ensuing grades.

Should numeration systems be an important topic in the elementaryschool mathematics program? Lois Lackner2 quotes a number of wellknown mathematics educators who strongly support the teaching ofvarious numeration systems. Such learning, they claim, would helpchildren understand the decimal system. Her complaint is that theseauthors wait until the fourth or fifth grade before introducing the topic.If it is so important, why not start this topic immediately? She writes,

So with such unanimity of opinion concerning the importance of teachingnondecimal and ancient systems of numeration to elementary school children, one maybe led to believe that we must introduce these topics with all due haste to our student onthe first day he enters elementary school (or even sooner). Already the child has beengrouping by tens for too long and supposedly not understanding the process. We mustbegin immediately to make clear to him that we use 10 digits, that we group by multiplesand powers of 10, that we can name numbers of any size using only 10 symbols, etc.3

Lackner suggests that experiences with other grouping be started asearly as kindergarten and she is convinced that such experiences would behelpful to children.

If the contention that experience with other numeration-systems helpschildren really understand the decimal system is a valid one, then, surely,experiences with the negative base systems should be even more effective.Much of the skills with other bases have become routine and almostautomatic. No one can add, subtract, multiply and divide in base nega-tive ten or base negative five without a real challenge to his understand-ing and ability. The negative bases offer a more profound understandingof place-value systems and thus give a better understanding of the deci-mal system.

2. Lois M. Lackner, "What About Numeration Systems at the Primary Level?" School Science and Mathematics,LXXLV (February 1974), pp. 152-156.

3. Ibid., p. 153.

Numeration Systems 29

Does the learning of other numeration systems enhance theunderstanding of the base ten? Research evidence has not beenconclusive. The majority of research studies that have been attemptedconclude that those children who learn only the base ten understand thedecimal system just as well, if not better, as those children exposed toother bases. Diedrich and Glennon,4 reviewing the studies that have beenmade, write:

First, it is safe to assume that a study of nondecimal systems will enhance the under-standing of the decimal system. Second, evidence is not conclusive that a study of non-decimal systems is more effective in achieving this objective than a study of base tenalone. Third, evidence does not tell us which method is more effective in promotingincreased understanding of processing decimal numerals, i.e., understanding therationale of computation. Fourth, evidence does not tell us which method is moreeffective in promoting increased understanding of a place-value system in general.Fifth, evidence does not tell us which method is more effective in promoting retentionof the understandings referred above.

They then conducted their own research study to determine if there isany conclusive evidence to support the contentions stated above. Theyconclude the following:

Relative to understanding the decimal system, and for the setting underconsideration, the obtained probabilities seemed to justify these conclusions: (1) Arelatively concentrated study of numeration is effective in accelerating the growth ofrelated concepts. (2) A study of bases three, five, six, ten and twelve is more effectivethan is a corresponding study of bases three, five, and ten. (3) A study of the decimalsystem alone is as effective as a corresponding study of nondecimal numeration. (4) Nosingle study is more effective than the others in promoting retention of achieved under-standings.5

Lackner’s article is based on a false premise, namely, that learningnondecimal numeration systems is essential for the understanding of thedecimal system. The study of nondecimal numeration systems has, infact, become a white elephant in the elementary school mathematicsprogram. The study of various bases was given a much greaterprominence than it deserved. If the time spent on numeration systemsmay be considered, to a large extent, to be a waste of time for mostchildren, such time waste is not in itself a great evil. The real evil effect ofthe overemphasis upon numeration systems is that it has createdconfusion in the minds of many teachers as to what mathematics is allabout. Too many teachers identify numeration systems with numbersystems. Not only teachers, but those who write for teachers, have thesame misunderstanding. An examination of the articles cited in the firstfootnote reveals that the phrase "number system" was used in almostevery article. The authors did not write about number systems. Theywere refering to numeration systems.

4. Richard C. Diedrich and Vincent J. Glennon, "The Effect of Studying Decimal and Nondecimal Numeration Sys-tems on Mathematical Understanding, Retention and Transfer," Journal For Research in Mathematics Education, 1(May 1970), pp. 163-164.

5. Ibid., p. 169.

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A numeration system is a language of symbols with which we representnumbers. A number system is a set of numbers, operations andpostulates. Each number system has its own properties. The naturalnumbers, integers, rational numbers, real numbers, complex numbers,etc., are examples of number systems. Each has its own peculiarproperties although several number systems may have similar properties.We can represent, for example, the set of whole numbers in base ten,five, two or twelve. All of the above number systems can be representedby any one of the numeration systems. The language of numerationsystems has little to do with the properties of numbers or with operationwith numbers. It is generally agreed that children should learn somethingabout the structures of various number systems. The emphasis uponbases, and certainly the identifying of numeration systems as numbersystems, has hindered the real understanding of number systems assystems. It would be better if the study of numeration systems wereeliminated from the elementary curriculum and the time thus saved weredevoted to the study of mathematics.Not only should numeration systems not be introduced in the primary

grades, as advocated by Lackner, but even in the higher grades shouldmany children never be exposed to nondecimal numeration systems.Should no children be taught other numeration systems? Not at all. Theteacher must be very careful about deciding what children could benefitfrom such study. A study of numeration systems does have a place forsome children in the elementary school program for several reasons. (1)It is a good cultural experience to learn that man can create varioussystems with which to communicate his ideas about number. (2)Although studies have shown that a comparison of the mean scores ontests for experimental and control groups shows no significant dif-ference, for some children in the composite score, there is a significantdifference. Some children may develop a better understanding of thebase ten because they were taught other bases. (3) For some children astudy of other numeration systems may be an exciting experience thatcould lead to further study of mathematics. In the words of Diedrich andGlennon,

None of the results of this study preclude the possibility that there exist reasons, otherthan those offered in this paper, for studying nondecimal numeration. Indeed, a fourth-grade study of nondecimal numeration might result in a rather remarkable shift inteacher-pupil attitudes toward mathematics in general�a perfectly legitimate reason forincluding a nondecimal study in the fourth-grade curriculum. A teacher should,however, be aware of the objectives to be achieved as a result of such instruction.6

6. Ibid.,p. 172.