FLM Publishing Association
Divisors and Quotients: Acknowledging Polysemy Author(s): Rina Zazkis Source: For the Learning of Mathematics, Vol. 18, No. 3 (Nov., 1998), pp. 27-30 Published by: FLM Publishing Association Stable URL: http://www.jstor.org/stable/40248276 Accessed: 01/06/2010 23:49 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=flm. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact
[email protected].
FLM Publishing Association is collaborating with JSTOR to digitize, preserve and extend access to For the Learning of Mathematics.
http://www.jstor.org
outrage and low risk-high outrage situations may help createthe credibilityto get the appropriatemessages across. Researchshould explore both aspects and may contribute insightsto informcurriculumplannersand teachers.
References AEC (AustralianEducationCouncil)(1991) A NationalStatementon Mathematics for Australian Schools, Carlton, Victoria, Curriculum Corporation. AEC (Australian Education Council) (1994) A Statement on Health and Physical Education for Australian Schools, Carlton, Victoria, CurriculumCorporation. Mann,R. andHarmoni,R. (1989) 'Adolescentdecision-making:the developmentof competence',Journal of Adolescence 12(3), 265-278. Pfeffer,C. R. (ed.) (1989) Suicide among Youth:Perspectiveson Riskand Prevention,Washington,DC, AmericanPsychiatricPress. Plant,M. andPlant,M. (1992) Risk Takers:Alcohol. Drugs, Sex and Youth. London,Routledge. Roscoe, B. and Kruger,T. L. (1990) 'AIDS: late adolescents' knowledge and its influenceon sexual behavior',Adolescence25(7), 39-48. Sandman,P. M. (1993) Respondingto CommunityOutrage:Strategiesfor Effective Risk Communication, Fairfax, VA, American Industrial HygieneAssociation. • Watson,J. M. (1997) Assessing statisticalliteracyusing tne media , m uai, I. and Garfield, J. B. (eds), The Assessment Challenge in Statistics Education, Amsterdam, IOS Press and The International Statistical Institute,pp. 107-121. Watson, J. M., Colhs, K. F. and Moritz, J. B. (1997) 'The development of chance measurement', Mathematics Education Research Journal 9(1), 60-82.
Divisors and Quotients: Acknowledging Polysemy RINA ZAZKIS Durkinand Shire (1991) discuss several differenttypes of lexical ambiguity,and suggest that by attendingto lexical ambiguity: we can identify the basis for particularmisinterpretation by pupils, and hence develop teaching strategies thatcircumventor exploit such tendencies,(p. 73) With respect to the mathematicsclassroom, they mention polysemy as one of the principalconcerns, namely certain wordshaving differentbut relatedmeanings.Wordssuch as 'combinations','similar', 'diagonal'or 'product'are examples of polysemous terms. The lexical ambiguity in these words is between their basic, everyday meanings and their meaningsin the 'mathematicsregister'of the language,that is, their specialized use in a mathematicalcontext. When a word means different things in different contexts, the intendedmeaningis usually specified by the context, including when differentmeanings of the word are presumedin everydaycontextand the mathematicsregister. When the intended meaning of the word in the mathematics registeris not availableto a student,a sense is often 'borrowed'from an everyday situation. The word 'diagonal', for example, used by a teacher in a geometry class, would probablyrefer to a segment connectingtwo vertices of a polygon. An insightful dialog about diagonals is pre-
sented by Pimm (1987, pp. 84-85), where a 13-year-old child interprets 'diagonal' as a "sloping side of a figure relativeto the naturalorientationof the page". Polysemy can also occur withinthe mathematicsregister itself, and context usually provides the primaryidentifier here as well. For example, the word 'operation'means one of the four - addition, subtraction,multiplication or division - in an elementary school classroom, while the same word means a function of two variables in a group theory course. We talk about the number 'zero' in elementary or middle school, and 'zeros of a polynomial' in an analysis course. 'Congruence'has different meanings in geometry and numbertheory. The meaning of a 'graph' depends on whetherone is thinkingaboutgraphingfunctionsin grade10 algebraor graphtheory.
'Divisor' and 'quotient'
1 would like to focus here on two instances of polysemous terms: 'divisor' and 'quotient'. In my view, they deserve special attention because the lexical ambiguity presented by these termsis not betweentheireverydayand mathematical usages, but arises within the mathematical context, within the mathematicsregister itself. In addition, as you will see, the context does not help in assigningmeaningsin this case. Both meaningsfor these words appearwithin the same 'sub-register'- the elementarymathematicsclassroom and a mathematicscourse for elementaryschool teachers. 'Divisor' has two meanings in the context of elementary arithmetic: (i) a divisor is the numberwe 'divide by'; (ii) from a perspectiveof introductorynumbertheory, for any two whole numbers a and b, where b is non-zero, b is a divisor (orfactor) of a if and only if thereexists a whole numberc such thatbe = a. The latter is a formal definition of a divisor in terms of multiplication.Formulatedin terms of division, b is a divisor of a in this sense if and only if the division of a by b resultsin a whole number,with no remainder. 'Quotient'means: (i) the resultof division; (ii) in the context of the division algorithm, the integralpartof this result.
Examples of discord 1: the classroom
Whatis the quotientin the division of 12 by 5? Therewas no consensus aboutits value among my studentsin a 'Foundations of mathematics' course for pre-service elementary teachers,andsuggestionsincluded2 (witha remainderof 2), 2 2/5, 2.4 and 12/5. Trying to determine the solution in a democratic way, we took a vote: 19 students voted for 2, 37 studentsvoted for 2.4, 2 2/5 or 12/5 (a combined count afteragreeingthatthese were essentiallydifferentrepresentations of the same number) and 9 students abstained. I purposefullyignored the lonely, uncertainvoice from the audience who claimed: "It matters what you mean by a quotient,doesn't it?". 27
However, democracy is not the best description of the regime in my classroom, so the majority vote was not accepted.The studentswere asked to reconsidertheir decision andto bringto theirnext class meetingjustificationsfor theirdecisions and to be preparedto defendthem. Rita was stronglyconvinced thatthe quotientin the divison of 12 by 5 was 2.4. She justified her decision by bringing her peers' attention to the lines from the course textbook(Gerber,1982): We say that6 dividedby 3 is equal to 2. The number3 is called the divisor,the number6 the dividendand the number2 the quotient,(p. 78) Based on this example, she concludedthatthe quotientwas whatyou got when you did division, and found furthervalidation for her conclusion in the RandomHouse Webster's College dictionary (1992), which defines a quotient as a "resultof division"(p. 1109). Wandaused the same textbook as the first reliable informationsource: Supposethat we have an arrayof a elements having b columns. Then the quotientof a divided by b, written a+b, is the numberof rows in the array,(p. 79) Arrangingthe elements in the following 5 column array, ***** ***** ** Wandawas convinced that 2 was the "only possible quotient".The following definitionon the subsequentpages of the textbookreassuredher: Let a and b be two whole numbers,b non-zero.Then b dividesa with remainderr (where0
