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EXPONENTIATION IS NOT REPEATED : DEVELOPING AS A CONTINUOUS

Arnon Avitzur e or University arnon.avitny.ed

The concept of exponents has been shown to be problematic for students, especially when expanding it from the domain of positive whole 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 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 rriclm and difficlties in nderstanding the rate of change of an eponential fnction and its . ne other problem that is rarely addressed in the 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

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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 Weber 2002 sggested that stdents be presented ith a description of an 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 binsy, 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 allos stdents to develop a comprehensive vie of eponents of positive ith 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 . 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 . 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 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

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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 , 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 . 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. oever, Davydovs frameor cannot be sed to analye the learning of the stdents from a cognitive perspective, and to complement it sed Simon et als 2010 frameor for learning throgh activity, bilding on Piagets 2001 concept of reflective abstraction. his frameor sees to eplain the transition from the point at hich a stdent did not nderstand something to the point at hich he or she does nderstand it p. 77. he folloing are three principles sed from Simon et als frameor the importance of a learners activity, the importance of the learners reflection, and the distinction beteen reflective abstractions and empirical learning processes p. 74. also note the goaldirected natre of the stdents activity, as ell as the development of the logical necessity that brings the stdent to anticipate the relevant reslts.

Principles in Developing the Concept of Exponents he development of a single image of eponentiation is the ey to developing a continuous concept of eponents, inclding natral and rational bases, as ell as real nmbers as eponents. he folloing are giding principles for designing a traectory that leads to sch concept Separating the Image and the Stdents shold be able to consider the calclation of any eponential epression separately from the image of eponentiation. Whereas the calclation of positive, negative and rational eponents may involve different operations, creating a single image of eponents ill allo them to reason abot those as one concept and not as a disconnected set, and carry the same image they epand their domain of operation.

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Beginning with a Qualitative Understanding of Exponents ne of the challenges in oring ith eponents is that the reslting vales of the operation icly become too big or too small. his does not allo stdents to develop a real sense of the rate of change since they need to maniplate large nmbers at the same time. eginning ith a alitative vs. antitative eperience of eponents that bilds on a visal representation ill allo the stdents to bild an nderstanding of the operations effects and develop an epectation for the reslt of sing different bases, ithot ever calclating the nmerical vale of the operation. Using Physical Quantities of a Continuous Nature n order to develop in stdents the nderstanding of eponents in the integer and rational domains, stdents shold not be limited to oring ith discrete antities. antities sch as length, , and volme allo them to perform the operation of eponentiation sing any vale for the eponent and eperience the change in a continos ay. Moreover, these antities are also measreable and visible so they lend themselves to the se of mltiple representations. Using Whole Numbers as a Case within the Continuous Domain eginning the calclations of eponents ith hole nmbers is a ay for stdents to find the reslt of the operation. he sggested approach introdces hole nmbers only once the image of eponents is established and ithot changing it moves to hole nmbers as a particlar case of calculating the reslt. Building on Students’ Intuitive Models Confrey and Smith sed a concept from the stdents orld that can be sed intitively as a basis for developing the mathematical concept, and it provides a strong fondation and an important entry point for the teaching traectory. Applying several of these principles together poses a great challenge. Continos models in natre e.g., temperatre change, decay of matter tend not to be directly measrable hich means they cannot be sed in a physical activity. he other models are mostly discrete similar to the ones shon in the splitting approach. An activity that ses a antity sch as length and reires the stdents to change its sie by, for eample, stretching it, does not reslt in an eponential change. n order to create a tre eponential change in a antity, the stdents have to consistently change the poerspeed they se hile stretching the antity, hich is not a natral thing to do. n order to overcome sch difficlties the sggested teaching traectory relies on a technological environment that allos the stdents to change the eponent continuously and observe the change in the antity.

What Is Exponentiation? he starting point for bilding a continos vie of eponentiation is accepting that repeated mltiplication is only a ay to calculate eponents in the case of hole nmbers, and not the image of it. his as nderstood by previos researchersConfrey and Smith, for eample, developed the image of eponentiation as a splitting operation and repeated mltiplication as a ay to calclate the nmber of elements after a seence of splits. ven thogh eponentiation is a mltiplicative idea at its core, this idea is not enogh for bilding a mental model that stdents can relate to as they epand the concept and learn to calclate it in different cases. ne of the main differences beteen addition, mltiplication and eponentiation is that the last one is an operation hich prodces a changing grothredction rate. t is also the basis for nderstanding the etreme changes that happen hen eponentiation is repeated. he basis for nderstanding the changing rate of eponentiation is that the grothredction in eponentiation is alays relative to the antity it operates on. ilding on that nderstanding, define the folloing image of eponentiation as the goal for the traectory

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Exponentiation is an operation that changes a given quantity multiplicatively based on the current size of the quantity. A fe notes ith regards to this image of eponentiation 1. hrogh defining eponentiation as a general changing operation, stdents are not limited to thining that eponentiation maes bigger hich is sometimes assmed from the definition of repeated mltiplication. he image can be sed in the cases of positive as ell as negative and fractional eponents. 2. he mltiplicative rate as the change factor allos for fractional rate bases, hich is important for bilding the single vie across bases that as the goal of the instrction. 3. here is an nderlying assmption, althogh not eplicit, that the stdents can thin of a change as related to a given sie, meaning a proportional change. 4. he estion of calclating the vale of the eponential epression is left ered in this definition. t is essential that the stdents have a single model to bild on, and that they see the calclations as based on specific cases ithin that image. Developing the image of eponentiation is a first step in helping the stdents to constrct the different cases of this image in the varios domains for bases and eponents. he image of eponentiation is reinforced by the se of symbolic langage in a more formal mathematical definition and model hich incldes the folloing components

1. 0 the antity being operated on, at the initial stage time0 2. a the mltiplicative rate of change natral or fractional in a given time nit 3. the change in time nits as related to the time of the initial stage

4. the reslting antity at time this can be at any point in time 5. a the accmlated mltiplicative change of the initial antity divided by 0.

Representation Model he se of mltiple representations and the tiliation of dynamic control and maniplation, made possible sing technology, can have a positive impact on stdents learning in algebra ieran, 2006 and these are incorporated as part of the representational model. n addition, the representation mst allo the se of continos antities and mae it possible for the stdents to engage ith the image ithot relying on specific calclations. The Context of the Problem ased on the desired characteristics of the image of the concept of eponentiation as described earlier, it is essential that the change described in the problem is one that is based on the crrent sie of the antity being changed. he sggested contet problem other variations eist of corse is the folloing Magic caterpillars need to eat a certain amount of leaves. The length (amount) that each caterpillar needs to eat is proportional to the caterpillar’s length. The eating transforms the caterpillar - they end up being as long as the length of the leaves they ate (for simplification they “eat” in straight lines). We will examine the changes in the length of the caterpillars. he contet as chosen to spport the development of a complete vie of eponentiation 1. t provides freedom in the selection of the initial antity to be operated on, as ell as the introdction of varios bases, in the form of different types of caterpillars that change differently, being more or less hngry 2. ts strctre allos for the introdction of both groth and decay, since the proportion of the food they need might be smaller or bigger than the caterpillars themselves 3. n the general form of the problem there is no mention of any period of time in hich the food needs to be consmed. t ill be introdced as a feeding cycle later as the stdents move to the antitative section in hich they calclate the vales of the groth. his also allos for the se .

of mltiplesie feeding cycles, hich is the basis for oring ith any rational eponents, and developing a continos vie of the operation The Visual Representation he representation incldes to critical components (a) Dynamic graphical representation of the caterpillar. Using a narro rectangle as a representation of the caterpillar allos the stdents to focs on the length as the relevant antity. Also, sing the length representation ansers the need for a continos antity, hich can gro or shrin dynamically. his eliminates the problem cased by sing discrete properties. (b) Horizontal time axis with a slide bar. he se of a slide bar represents a continos vie of the eponent vale alloing the stdents to or in fractional and integer vales. ffering a dynamic maniplation of one element, the se of a slide bar is a non representation for time progression hich is familiar to stdents consider obe for eample. ringing all the pieces together, and sing the elements of the mathematical model ith the representation model and the problem contet, e have the folloing

1. 0 is the sie of the caterpillar at the given initial state hen the observation begins 2. a is the property of the caterpillar hich defines ho mch food it eat at each time nit 3. is the change in time represented by the slide bar. he location of the marer on the slide bar shos ho far they are and in hich direction from the initial state

4. is the sie of the caterpillar the rectanglar bar at any point in time the slide bar 5. a is the reslt of calclating or predicting the change from the initial to the crrent state

Analysis of the Hypothetical Learning he affordances of the definition and model described above are best demonstrated throgh an analysis of a stdents hypothetical learning process. What follos is a description of the hypothesized learning of a stdent ho performs to activities from the first tas in a teaching traectory hich is based on the principles above. A fll description of the teaching traectory and analysis of the hypothesied learning as for each step is beyond the scope of this paper. D the first three activities of the traectory not detailed here, the stdents or ith antities the length property of the caterpillars, ith no particlar nmerical vale bt of a comparable magnitde that change in either eponential or linear form and nderstand that eponential groth or decay is faster than linear one, once the antity reaches a certain sie. hey can eplain the relationship beteen the sie of the caterpillar and its groth or decay and focs on the fact that the bigger the given antity is, the bigger the change is. Also, stdents are introdced to the formal . hey no move to the forth activity. Activity 4: Comparing Different Quantities n this activity, the stdent ors ith to caterpillars ith the same base for the eponent and compares their groth, in absolte additive and proportional terms. Woring in the same base and the same time period ith differentsied caterpillars focses the stdent on the initial sie of the caterpillars. n reflecting on his activity he is epected to nderstand the logical necessity of the initial antity eplaining the difference in the reslting antity, bilding on his prior noledge of the relationship beteen the antity and the change, and noing that the initial antity is the only attribte in hich the caterpillars vary. describe no the steps in the activity that lead to this nderstanding. n the first step of this process, the stdent ors ith one caterpillar and establishes its groth rate, as in previosly activities. e does this by sing the slide bar mared ith nits as before to change the time vale and compare the reslting sie. From this, he can see that the groth is based on a particlar base rate. For eample, he might observe that the caterpillar gros by a mltiple of 3 for every time nit in the case of a base of 3. e does this by comparing the antity after one time nit, ith the antity at the beginning, or the antity after 1 nits, ith that of after nits. .

nce the groth rate is determined, another antity of a bigger sie is introdced and placed net to the original one. he stdent is told it needs to eat the same proportion as the previos one, and is ased to predict hich one ill gro more. he stdent is epected to develop an nderstanding that the groth of a caterpillar is proportional to its original antity, and is assmed to no that hen mltiplying to nmbers by the same factor, the larger nmber ill reslt in a larger prodct. ilding on these to, the stdent ill anticipate that the larger caterpillar ill gro more, becase 3 times a bigger nmber is bigger. his nderstanding is poerfl since the stdent learns abot a relationship hich is not dependant on hether the given antity is of a hole or fractional sie. At the end of this activity the stdent concldes that for an eal amont of time nits and the same groth rate, the larger antity ill gro by a larger amont. t is important to note that the stdent generalies abot eponents without looing for a nmerical pattern, and that the process he goes throgh is based on reflection abot the general process and the nderstandings developed abot the natre of this activity. Activity 5: Moving Forward and Backward in Time, Only Until the Present n this activity stdents can move the scroll bar forard and bacard beteen the present time and a particlar point in the ftre st to avoid infinite grothdecay in the ftre. hey begin ith a antity of a particlar sie and are ased to eplain hether it is groing eating more than its sie or shrining eating less than its sie, based on their observation of the behavior. his shold be a simple estion for the stdents, noing they completed the previos activities in hich they learned that hen a caterpillar eats more than its sie, it gros every day toards the ftre and the opposite in decay. he stdents are then ased to eplain hether the antity gros bigger or smaller if yo move forard in time. he stdents are then ased to move the bar to a random point on the time line and anser the folloing estion in order to reach this state and from hat yo no of the behavior of the caterpillar, old the antity had to be bigger or smaller before this point in time. Stdents, bilding on their forard thining and activity of moving the bar forard from before, see the logical necessity that in order to reach a certain antity, hen a caterpillar is groing, it had to be smaller before and similarly for decay it had to be bigger before. hey are then ased noing that a caterpillar is groing caterpillar, if yo loo bac in time, old that caterpillar be bigger or smaller and ho old it loo if yo move forard in time Althogh this can be checed by the stdents throgh the representation, at this point they already see the necessity of the caterpillar being smaller, since it had to gro to reach the given sie in the case of groth. he stdents learn to anticipate that if the caterpillar is of a groth type a1, then hen moving forard in time increasing the eponent the antity gros, and hen moving bacard in time decreasing the eponent it becomes smaller. At the end of those five activities, stdents develop the concept of eponentiation as a change hich has a rate proportional to the crrent sie of the antity, as a factor of time. Also, stdents can eplain the change in a antity hen moving both forard and bacard in time, hich ill serve as a precrsor for the development of ero and negative eponents as places on the time line. Moreover, the se of a continos time line sets the stage for other time points hich ill not be hole nmbers. Stdents never sed particlar vales for antities so they never calclated the eponential change, and that spports the development of a continos vie of eponentiation and eeping the image of eponentiation separate from the calclation process.

Developing Continuous Concepts in Mathematics he development of mathematical operations sch as addition, mltiplication and eponentiation sally begins ith the positive hole nmber case and epands later to negative and rational nmbers. began this paper presenting the implications sch an approach might have on the stdent and sggested principles for designing a teaching seence for the development of one of those concepts in a continos manner. Althogh focsed on eponentiation, the importance of the or might be beyond a particlar content area, and perhaps also it can serve as an eample of ho other continos concepts might be thoght of in sch fields as calcls and operations in algebra.

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Moreover, the approach laid ot here might also serve as basis for designing the development of continos concepts in general. he vale of sing to different theoretical frameors is revealed throgh the eamination of the hypothetical learning seence in hich the stdent on the one hand begins ith the general image of eponentiation, as sggested by Davydov, coming from a sociocltral perspective, bt at the same time, develops the nderstanding hich is eplained from a constrctivist perspective. he se of one frameor as the leading one for the overall design of the seence and another frameor to design the activities ithin the seence might have applications in other conceptal .

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