EXPONENTIATION IS NOT REPEATED MULTIPLICATION: DEVELOPING EXPONENTIATION AS A CONTINUOUS OPERATION Arnon Avitzur e or University arnon.avitrny.ed The concept of exponents has been shown to be problematic for students, especially when expanding it from the domain of positive whole numbers to that of exponents that are negative and later rational. This paper presents a theoretical analysis of the concept of exponentiation as a continuous operation and examines the deficiencies of existing approaches to teaching it. Two complementary theoretical frameworks are used to suggest an alternative definition for exponentiation and guiding principles for the development of a teaching trajectory, and then to analyze an example of the hypothetical learning of a student who goes through the first task in the trajectory. The paper concludes with some possible implications on curriculum and task design, as well as on the development of mathematical operations. eyords Algebra and Algebraic hining nstrctional Activities and Practices The Problem with Exponents esearch has fond the concept of eponents problematic for both stdents and teachers of all levels Confrey Smith, 1994 lsta, 2007 Goldin erscovics, 1991 Weber, 2002. he most common definition that stdents have for eponents is that of repeated mltiplication, for eample 25 2·2·2·2·2 hich is to mltiplied by itself five times. his limited vie of eponentiation, althogh simple to nderstand, prevents stdents from nderstanding the behavior of eponents in nonnatralnmber poers. For eample, if eponentiation is repeated mltiplication then something to the poer of ero might be seen as ambigosshold it be ero or one Moreover, there is no meaning to mltiplying something by itself a negative nmber of times. An etension to fractional poers that preserves the sense of repeated mltiplication is an impossible tas, and often fractional eponents are presented as a different ay to rite radicals and connected to repeated mltiplication in an artificial ay. As they finish their high school nit on eponents, stdent end p not being able to loo at eponentiation as a continuous process in terms of the eponent vale and, in the best case scenario, have a fe different models connected to one another throgh loose logic. Althogh it is possible to develop the concept of eponentiation as a casebased operation, this approach may reslt in varios negative implications, sch as the perception that eponentiation is alays an increasing operation, an inability to or ith eponents as continos fnctions later in the crriclm and difficlties in nderstanding the rate of change of an eponential fnction and its derivative. ne other problem that is rarely addressed in the mathematics edcation literatre lies in the fact that stdents do not have any alitative sense of the changing groth rate of an eponential fnction that reslts in an inability to eplain sch ideas as componding or poplation groth ithot having to calclate its vale nmerically. Existing Approaches here have been cases of teachers ho tried to develop stdents nderstanding of the eponents rles throgh the process of proof and mathematical consistency by moving from one rle to another in dedctive manner ith the goal of alloing the stdents to see the connection beteen them. his, hoever, does not create a single vie of eponents that stdents can or ith across domains, and it alays remains as a seence of logical operations that eplains the varios cases of eponents positive, negative, ero, and rational. he research commnity has made several attempts at developing a . conceptal nderstanding of the concept of eponents hile at the same time aiming at bilding a single vie of the operation across domains. The Functional Approach he teaching of algebra in a fnctionalbased approach as first sggested by Goldin and erscovics 1991, later to be tested by lsta 2007 in a teaching eperiment. n this approach, the nderstanding of negative, ero and rational eponents comes from constrcting the definition of an eponential fnction starting from natral eponents, and later investigating this fnction to epand the notion to other domains. n lstas teaching eperiment, some of his stdents ere able to logically connect the different cases of eponents, bt cold not give a single definition for all of them. Developing Algorithms Weber 2002 sggested that stdents be presented ith a description of an algorithm to compte eponents, hich later they epress formally. he stdents rote the eponential epressions as prodcts of factors, and then completed activities in hich they debated abot the natre of rational eponents. Althogh Weber eplained the epected learning path sing APS theory Dbinsy, 1991, it as not clear ho the debate stage helped the stdents develop a conceptal nderstanding of rational eponents that aligned ith their original definition of the operation. Exponents as Splitting Confrey and Smith 1994, 1995 sggested that eponents be developed throgh the idea of splitting. asing their design on stdents familiarity ith the idea of fair share and splitting, they gave special attention to the rate of change of the fnction, hich as not given by other researchers. hey aimed at developing in stdents the mltiplicative comparison beteen the sies of the antity at different stages, ith a focs on a mltiplicative rate of change as being fied throghot the or ith eponents, in contrast to the varying additive rate of change. his basis allos stdents to develop a comprehensive vie of eponents of positive base ith integer poers positive, negative, and ero. o etend the domain to rational bases and eponents, they relied on the contrast beteen the conting and the splitting orlds and offered a logical eplanation that acconts for rational eponents. n contrast to their initial or that relied on splitting, the epansion to rational eponents does not offer a cognitively intitive model to or ith, and the casebased vie is left nresolved. What Is Missing? Althogh some of those stdies contain valable insights, each has its on drabacs ith regards to the development of a comprehensive conceptaliation of eponentiation as a continos operation relying on a single image. ne shared limitation of all of the above approaches is that they begin oring in natral nmbers and give the stdents a limited concept of eponents, later to be epanded in one ay or another. pansion to the real nmber domain is accomplished throgh formal eplanations, definition or logical reasoning, bt is not based on the original image the stdents have. ven in the splitting orld, hich is an etremely poerfl idea based on stdents on eperience, there is no real meaning to the eponent itself that acconts for both positive and negative vales. t ends p being a logical process eplaining to the stdents that in the ne mltiplicative orld negative eponents reslt in division. ational poers are similarly not ell addressed in the splitting orld ending p being nderstood empirically throgh the identification of patterns. t is etremely hard to thin of doing three and a half split operations since the splitting is done on contable sets. n essence, trying to bild stdents nderstanding based on a limited domain natral nmber as a starting point reires that stdents adst their definition of eponents for every ne domain etension, sometimes conflicting ith the original one they had, reslting in a disconnected set of definitions. Another limitation of the approaches is an overse of calclations as a ay to nderstand eponents, indicating a move toards an empirical instead of conceptal nderstanding. his is evident in the . algorithmic and fnctional approaches, and even in the or based on splitting, there is still some emphasis on generating sample vales in order to nderstand rate of change. Theoretical Frameworks ne goal hen designing the folloing teaching traectory is that it ill allo stdents to develop a coherent nderstanding of the concept of eponentiation as a continos operation ith no contradictions as the domain of the eponent epands from positive hole nmbers, to negative hole nmbers and ero, to fractional eponents, and finally to all real nmbers. n order to design sch a traectory and to analye its seflness, am sing to different theoretical frameors, hich arge to be complementary Simon, 2009, and offer principles for the arrangement of the content and provide constrcts that eplain the learning of the stdent. For the overall content organiation approach and the analysis of the core nderstandings that are the fondation of the concept of eponents, sed the design principles as laid ot by Davydov 1975. ne of Davydovs main points is that the material shold be organied, moving from the general to the specific. hat is, the stdents shold move from learning abot the concept in its most general form and later to or throgh different cases hich are manifestations of it in sch a ay that they develop a complete nderstanding of the concept. A principal goal in designing the teaching traectory as to develop an image of eponents that stdents can repeatedly se in different cases ithot ever having to deal ith contradictions. his is significantly different than bilding on the specific case of positive hole nmbers and later epanding it to negative and fractional eponents. A giding eample of the development of a concept from general to specific is the development of mltiplication Davydov, 1992. Davydov asserted that mltiplication shold not be vieed as repeated addition, raising similar isses to the ones described earlier ith eponentiation being vieed as repeated mltiplication. n his measrementbased approach, he distingished beteen the ay one calculates the vale of a mltiplication epression and the image of mltiplication. Creating an image of mltiplication as a change of nit of measrement allos Davydov to develop ith his stdents a more general vie of mltiplication that also incldes mltiplication of fractions. My or bilds on the previos or not becase it is intended to address the isse of teaching mltiplication, bt becase this eample is sed to demonstrate ho it is possible to create a single image of a concept that can be sed across cases.