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Home , Mathematical proof

RELATIONSHIPS BETWEEN MATHEMATICAL AND

David Plaxco irginia ech dplacovt.ed

This article discusses results from interviews with two undergraduate students in an introductory proofs course. The researcher assessed the participants’ general proof schemes and built models of the participants’ conception of probabilistic independence and mutual exclusivity. The participants were then tasked with asserting a relationship between independence and mutual exclusivity and trying to prove the asserted relationship. The results discuss possible interactions between students’ conception of mathematical ideas and their approaches to proof.

eyords Proof Scheme Definition Concept mage

Mathematical proof and mathematical definition are to areas of research that have recently gained heightened attention from researchers dards Ward, 2004 arel Soder, 1998 o, 2010 inner, 1991. hese to areas, thogh generally stdied separately, are intrinsically related dards Ward, 2004 Weber Alcoc, 2004. ittle research, hoever, has eplicitly eplored relationships beteen proof and definition. he prpose of this research is to eplore the connections and relationships beteen ndergradate stdents proof schemes and their nderstanding and se of definition.

Theoretical Framework A Framework for Discussing Proof arel and Soder 1998, 2007 have provided a fndamental frameor for research in stdents conceptions of proof. his frameor begins by defining proof as a topart process ascertaining and convincing. A stdents notion of hat constittes ascertaining and persading is called that stdents proof scheme arel Soder, 1998, p. 244. here are varying levels of proof schemes described in the literatre. roadly, these proof schemes are ternal, mpirical, and Analytical an ternal proof scheme incldes validations by athority, rital, or symbolism, an mpirical proof scheme is indctive or perceptal, and an Analytical proof scheme is more rigoros and logical. t is important to note that, at a given time, no person completely displays of eactly one proof scheme. ecase of this, a stdents proof scheme is a generaliation of the types of proof schemes evident throgh his or her or. For eample, a stdent cold ehibit both mpirical and Analytical proof schemes ithin a short period of time or even ithin a single proof sch a stdent cold be said to have an emerging Analytical or mpiricalAnalytical proof scheme. Weber and Alcoc 2004 also contribte a frameor for discssing stdents semantic and syntactic proof prodctions. his frameor dras distinctions beteen stdents se of instantiations in proof syntactic and the formal maniplation of logical mathematical statements semantic. While this frameor is constrcted otside of arel and Soders 1998 proof schemes, the to classification systems for stdents mathematical tendencies seem as thogh they cold or sccessflly together as neither ecldes the other. A Framework for Discussing Definition inner 1991 distingishes beteen the ideas of concept image and concept definition. e defines the concept image as a nonverbal entity sch as a visal representation of the concept or a collection of impressions or eperiences p. 68 hich or mind associates ith the concept. n contrast, the concept definition is the formal mathematical definition of a concept. hese to ideas are not necessarilyand, one cold arge, seldomthe same thing. inner ses the sentence, my nice green car is pared in front of my hose, as an eample of concept image p. 67. e arges that the reader or listener does not

necessarily consider the definition of each ord in the sentence, bt that each ord invoes a generic concept image in his or her mind, the collection of hich allos the sentence to tae form as a hole impression. Probability as a Context Manage and Scariano 2010 provided a sefl contet in hich this research as condcted. he athors fond that most of the stdents in their stdy thoght that to events being independent implied that they ere mtally eclsive and vice versa. Althogh ones initial reaction may be to conclde this eact relationship, after carefl consideration of the to concepts one realies the to terms have almost eactly opposite meanings. his almost is attribted to cases in hich one or both of the events has ero probability. therise i.e., if to events have nonero probability, independence implies that to events are not mtally eclsive and mtally eclsive events are not independent. When ased to prove this relationship beteen independence and mtal eclsivity, one mst address his or her on conceptions of independence and mtal eclsivity, compare the to concepts, ascertain the relationship beteen the to, and try to convince others. So, this relationship beteen mtal eclsivity and independence ill provide a contet for eploring the se of definition in proof.

Methods he researcher condcted semistrctred intervies ith three ndergradate stdentsAle, etty, and Carolineho ere enrolled in an ntrodction to Proofs corse. All three stdents ere maors in their third year and ere chosen randomly from a grop of volnteers. one of the stdents ere compensated for their participation. ach participant completed three intervies, each lasting approimately one hor. ach of the three intervies had its on nie goal ntervie 1 to gage the participants general proof schemes, ntervie 2 to gain insight into the participants concept and concept images of specific mathematical terms, and ntervie 3 to observe and analye the participants se of definition and imagery hile proving relationships abot the mathematical terms discssed in the previos intervie. Collection All intervies ere recorded sing both video and adio devices. he researcher ept notes throghot the intervies and all participant or as collected. he first intervie as designed to gather a general nderstanding of each participants proof scheme. he intervie consisted of each participant assessing a matri of proofs, hich is a 3by3 grid of mathematical proofs. ach ro in the matri contained three variations of proof of the same mathematical relationship, reflecting arel and Soders 1998 three maor proof schemesAnalytical, ternal, and mpirical. he participants ere ased to assess each proof for mathematical and logical correctness. From these responses, the researcher determined the aspects of mathematical proof that the participants considered important andor necessary or, conversely, nimportant andor nnecessary. n trn, the researcher sed the participants responses and reasoning in order to form a notion of each participants general proof schemes. n the second intervie, the researcher collected the participants definitions of mtally eclsive events and independence. he researcher also ased the participants to consider several events in varios sample spaces and determine hether pairs of events ere mtally eclsive andor independent. Participants ere also invited to introdce their on events and sample spaces to elaborate points that came p dring discssion. his as intended to provide the intervieer ith insight not only into ho the participants defined each of the to terms, bt also ho these terms ere applied in varios probabilistic contets. he researcher cold then distingish beteen the participants concept definitions collected directly and concept images dran from eamples, phrasing, etc.. he third intervie as designed to provide a contet herein the participants cold assert a mathematical relationship beteen independence and mtal eclsivity and then attempt to prove this relationship. he participants ere ased to assert to main relationships given that to events have nonero in the same sample space, does independence imply mtal eclsivity and does

mtal eclsivity imply independence hese estions ere posed as to separate mltiplechoice estions, as in Manage and Scariano 2010. he researcher analyed each video and adio recording after each intervie in order to determine the participants proof schemes, identify maor themes in the participants reasoning, model participants nderstanding of mtal eclsivity and independence, dra otes from the dialoge to spport sch models, and develop individalied clarifying tass for the sbseent intervie. hroghot the video analysis, video clips ere taen that spported or challenged oring models of participant thining. hese videos ere collectively reanalyed in order to confirm or reevalate a model. he researcher old then develop estions for the sbseent intervie that old be sed to help clarify conflicts ithin the model.

Results For the sae of depth and limited length, e discss only Ale and etty here. Alex hroghot the first intervie, Ale ehibited a predominantly Analytic proof scheme. ventally he correctly spported all Dedctive proofs and refted all mpirical and ternal proofs, citing appropriate flas in or reasoning. n a fe instances, he shoed signs of relying on a proofs form rather than content, signifying an occasional tendency toard a ital ternal proof scheme. Ale as also very pedantic abot precise details, reflecting a septical point of vie and checing for logical progression in each proof. Ale displayed a deep nderstanding of eamples and their se in proof. his as icly evident in the first eample in the matri of proofs. his proof, applying an ndctive mpirical proof scheme, sed an eample of a large random nmber that ehibited the desired reslt. After reading the argment, Ale immediately said, eah, this is bogs. e later refted other indctive proofs very similar to the first. hese eamples highlight Ales ability to refte proofs that inappropriately se eamples. Another interesting aspect of Ales proof scheme is his emphasis on the aioms of real nmbers and considering the space in hich he as oring. hese alities ere evident in three instances. n the first to cases, Ale eplicitly applied the aiom of the closre of nder addition and mltiplication. n the second case, Ale also invoed the associativity aiom for real nmbers. n the third case, Ale sggested that the sm of the interior angles of a triangle might not be 180 in nonclidean space. While this cold be a manifestation of the rigor reired in his ntrodction to Proofs corse, it is evident from these eamples that Ale had internalied a mindset that considers the system in hich a proof is arged and its fndamental aioms. t shold be noted that Ales se of aioms in this intervie reflects arel and Soders 1998 ntitiveAiomatic proof scheme. hese alities of Ales proof scheme combine to spport an initial emphasis on the form of a proof and then carefl investigation of motivation and stification at each line of an argment. his emphasis as manifested in his pedantic discssions of the proof riters stification, his se of aioms, and reftation of proofs by eample. We see Ale sed the form of a proof to mae initial dgments, bt his septicism forced him to evalate a proof based on its linebyline merit. From this, e can conclde that Ale generally displays an ntitiveAiomatic Analytical proof scheme ith tendencies toard a ital ternal proof scheme. n the second intervie Ale eplored to sample spaces and discssed a fe other eamples that he sed to help describe his nderstanding of mtal eclsivity and independence. As e ill see, Ale displayed an etremely internalied and poerfl conception of independence. Ale defined independence as, hen the otcome of one event does not affect the otcome of a sbseent event. his definition implies an emphasis on a seence of events, here one of the events being considered mst occr prior to the other.

With regard to mtal eclsivity, hoever, Ale as less certain of a formal definitionchanging his definition tice throghot the intervie ntil eventally declaring, Mtal eclsivity is hen performing an event or series of events cases a sbseent event to have ero probability of happening. Again, Ale ses the ord sbseent in his definition, hich implies that this relationship is defined over a period of time. t is important to note that Ales initial definition of mtal eclsivity consistent ith the mathematical definition as not defined over time, bt rather instantaneosly. t as not ntil he had considered eamples in the to given sample spaces that he changed this definition to more closely resemble his definition of independence. When prompted for an eample of independent events, Ale gave to eamples a die and a coin. e stated that rolling a si on the first roll of a die does not affect rolling a si on the second roll of a die and gave an analogos eplanation for the coin. hese eamples are consistent ith his definition of independence, implying that the to events in consideration tae place at separate times. is initial eamples of mtally eclsive events ehibited hat he described as elldefined states inclding raining verss not raining, sides of a die yo cant roll both a 5 and a 1, and a coin its either heads or tails. hese eamples spport his original definition, hich considers the to otcomes instantaneosly in that it cannot both rain and not rain at the same time. ater in the intervie, hoever after changing his definition of mtal eclsivity, Ale described repeatedly draing any card ithot replacement ntil all spades ere ehasted. n this case, draing any card and draing a spade ere mtally eclsive since draing any card can eventally case draing a spade to have probability ero. his eample seems mch more convolted than the first three eamples, bt spports Ales neer definition of mtal eclsivity. We can see that Ales conception of independence as so strong that it not only inflenced ho he defined mtal eclsivity, bt also cased him to reect three different eamples and develop a ne concept image for mtal eclsivity herein one event mst case a sbseent event to be impossible. his ne concept image as so strong that, hen ased to reconcile this ne definition ith his original eamples, Ale reneged on their mtal eclsivity e.g., heads on a coin does not case not tails later. ally intriging is the fact that Ale independently asserted a corollary to his ne definition of mtal eclsivity. n this corollary, Ale stated that if the to events are mtally eclsive, then they cannot be independent. his reflects the almost eact relationship otlined in Manage and Scariano 2010 and investigated in the third intervie of this research. Ale sed an eplanation analogos to that described in Manage and Scariano 2010. e asserted that, since one event cases the second event to have ero probability, the first event changes the probability of the second event and therefore the to events are not independent. t shold be noted, hoever, that Ale did not consider the case hen the second event in the seence already had ero probability. n the third intervie, Ale responded that if to events ere mtally eclsive this implied that they ere not independent. his claim as made sing his final definition of mtal eclsivity. e directly referenced his on corollary from the second intervie in hich he made this eact assertion. Ale also claimed that if to events are independent then they are not mtally eclsive. e spports his anser choice by saying, one events not affecting the other event at all so, mean, its not going to case it to have ero probability case its not changing the probability of the net event. As ith the first estion, this anser choice spports the relationship beteen the mathematical definitions of independence and mtal eclsivity for nonero events. Betty etty displayed a predominantly Analytic proof scheme ith the eception that she accepted one proof based on its appearance and another proof based on its form. etty correctly refted the three eamples of ndctive mpirical proofs, bt accepted one Dedctive proof becase it seemed more mathematical. er reftation of the indctive proofs shos her nderstanding of the importance of a general proof for all cases. ettys acceptance of a proof based on its seeming mathematical alities and acceptance of a false proof by the principle of mathematical indction, hoever, indicate a tendency toard ternal ital and mpirical Perceptal proof schemes.

etty shoed an insistence on nderstanding very specific aspects of a proof rather than draing any assmptions abot the proofs process ith the eception of one case. She icly accepted a proof by mathematical indction. ere, she as liely preoccpied ith the form or loo of the proof, rather than its mathematical . his idea as spported hen etty stated that her had recently discssed the principle of mathematical indction. When ased hich of the first three proofs she preferred, etty chose the last proof becase the processes in the second proof ere not obvios to her. his reflects a need to nderstand connections in a proof, even thogh this need as temporarily sspended in the case of mathematical indction. his need as also addressed later in the intervie, hen etty described the process of verifying for herself relationships she felt she did not nderstand in class. hroghot the rest of the first intervie, etty correctly refted mpirical and ternal proofs and accepted Analytical proofs. She reected the false proofs ith little hesitance. At one point, etty described combinations of negating the hypotheses statements of the ndctive proofs, shoing a clear nderstanding of logical proof, contereamples, and proof by . She also reflected an ability to identify false proof by eample. hese eamples sho a healthy septicism of Athoritarian and ital proof, both of hich are ternal proof schemes. Additionally, ettys eplanations in refting ndctive proof schemes spport an emphasis on proof for all cases. When ased hat it meant for to events in a sample space to be independent, etty responded, he intersection is ero. s it hats hat m asing. dont remember. etty almost instantly changed this to, o events are independent if the probability of A occrring does not affect the probability of occrring. etty then described the independence of events A and sing the eation PA PA. either of these representations necessarily implies a chronological distinction beteen events A and as as seen ith Ales se of the ord sbseent. t, hen prompted for an eample of independent events, etty described the act of picing a card from a dec of fiftyto cards and ptting it bac so that the probability of picing a second card is not affected. Similarly, hen ased for an eample of events not being independent, etty provided the case of picing a card and not replacing it. hese eamples are consistent ith a conception of independence in the contet of a ith replacement and ithot replacement conditioning event. n contrast, etty defined mtally eclsive events ith the , yo cant have both at the same time. his definition eplicitly states that the events can be compared instantaneosly. ere, etty gave the eample that the choosing the een of hearts and choosing the ac of diamonds are mtally eclsive becase they cannot both occr hen one card is dran. We notice that this definition is consistent ith the mathematical definition and that this eample is consistent ith ettys definition. etty did spend mch more time defining mtally eclsive events compared to her definition of independence, bt once she determined this definition, she held firm to its accracy saying, m sorted no. his reflects her need in first intervie to prove relationships in order to nderstand them. ettys initial confsion of independent events as events that dont happen at the same time reflects the most common misconception in Manage and Scariano 2010. Althogh she icly changed her mind abot the definition of independence, this confsion as apparent in her se of mathematical notation to represent the ideas discssed belo. Also, hen eplaining her conditional notation of independence, etty described to independent events as completely separate, hich one cold arge is a descriptor more applicable to mtally eclsive events since their intersection is empty. More than once, etty rote an eation involving probabilities saying, hats st something remember from probability. For instance, she initially sed PA0 to represent independence and sed the eation PAPAP to define mtal eclsivity. hese eations ere icly erased. he former, hoever, as eventally sed to describe mtal eclsivity. For the latter, etty admitted, have no idea here that came from or if thats even mtally eclsive. And old not be able to come p ith it. We notice that ettys second of independence, PAPA, is tre nless the probability of is ero. n this case, the statement PA maes no sense, althogh it cold be adapted to say, to events A and are independent if both have nonero probability, PAPA, and PPA.

n this intervie, e see that ettys concept definitions, thogh initially inconsistent, are each strongly internalied hen evalating the independence and mtal eclsivity of specific events in specific sample spaces. his is evident becase once etty defined each term, she as sorted on ho to verify them and seemed to develop ic checing schema in order to do this e.g., Can these happen at the same time. er spoen reasoning for to events independence and mtal eclsivity reflected these ic checs. n response to each of the to estions in the third intervie, etty conclded that there as not enogh abot the sample space and that to mtally eclsive events can be both independent and not independent. his led her to respond that there as not enogh information abot the sample space or the contet of selecting events in the sample space to determine a relationship. She eplained that in the previos intervie she had seen mtally eclsive events that ere both independent and not independent a copy of her responses from the second intervie as presented to her. She also eplained that she sa independent events that ere both mtally eclsive and not mtally eclsive in the second sample space. From this, etty reasoned that more information as needed abot both the sample space and the actions taen beteen the occrrence of the first event and second e.g., replacement, nonreplacement. Again, e see independence is affected by the contet in hich the events tae place. ettys proof scheme shoed that she is more inclined to ant to verify mathematical relationships on her on. his as evident as she sorted herself abot the definitions of independence and mtal eclsivity. Dring this process, etty sccessflly reconciled her definitions of the terms ith symbolic representations abot hich she as admittedly nsre that she had recalled from her corse. etty sed these definitions to investigate the sample spaces in the second intervie, the reslts of hich had a direct affect on her reasoning in the third intervie. ecase ettys definition of independence relied so heavily on the sample space and hether replacement occrred, she had eamples of all different combinations of independence and mtal eclsivity.

Conclusion We see that proof schemes can be both restricted and enhanced by stdents definitions of the mathematical ideas they consider. hogh her reasoning as logically based on her previos eperiences in the samples spaces, ettys conception of independence and mtal eclsivity cased her to reire more information abot the sample spaces in estion, in trn restricting her ability to dra conclsions beteen the to concepts. n the other hand, Ales ability to adapt his concept image and concept definition of mtal eclsivity alloed him to logically conclde both directions of the relationship beteen mtal eclsivity and independence, hoever correct or incorrect his definition may have been. n his proof, Ale claimed from his concept definition of mtal eclsivity that each mtally eclsive event old case the other to have ero probability. his old mae the to events not independent since his definition of independence necessitated each event to not affect a sbseent event. Using similar reasoning, Ale conclded that independence implied not mtal eclsivity. t shold be noted hoever that, despite Ales focs on proof for every case in the first intervie, he failed to assert a relationship for the case hen one or both events ere given to have ero probabilities. he contrast beteen his assertions abot proof and his actions in proving this relationship reflects the pathological natre of ero probability cases pointed ot by elly and iers 1986. nterestingly, this also points to a characteristic of his definitions that may have inflenced his thoght process an event ith ero probability cannot happen first and therefore can neither case nor affect any other event, as the definitions reire. ettys as nable to logically assert any certain relationship beteen the to concepts. his reslted from sch strong concept images of independence and mtal eclsivity. More specifically, ettys personal eperiences in the sample spaces alloed her to provide contereamples to any eplicit relationship beteen the to concepts. Since specific characteristics of sample spaces affected to events independence, she reired information abot a sample space in order to mae abot the events in estion. his prevented etty from generaliing to all cases an eplicit relationship beteen mtal eclsivity and independence, hich her proof scheme reired.

ecalling the Ale and ettys general proof schemes mostly Analytical ith slight mpirical and ternal tendencies, e consider ho these related to their se of definition. Ales dynamic concept image and nsolicited prodction of the lemma for the definition of mtal eclsivity reflect an Analytical frame of mind that is also geared toard finding and asserting relationships beteen the to concepts. We see ith etty, hoever, that a mostly Analytical proof scheme alone is not sfficient to connect the relationships beteen mtal eclsivity and independence. his is becase her conceptions of the to ideas ere so poerfl that she as comfortable sing the for cases from her eploration to sho that no relationship eisted. From these to cases, e see that little can be made abot ho a stdent ses definition relative to arel and Soders 1998 proof schemes. t e can also consider these cases ith respect to Weber and Alcocs 2004 semantic and syntactic proof prodctions. ecase he prodced it immediately after changing his concept definition of mtal eclsivity to more closely resemble his concept definition of independence, e see that Ales lemma and therefore responses in the third intervie as a direct reslt of his comparing the to concept definitions. A syntactic approach to the relationship as not fritfl, hoever, ntil he changed his definition. Conversely, ettys se of previos instantiations a semantic approach prevented a definite relationship beteen the concepts from forming. t is nclear, thogh, hether etty even thoght her concept definitions might need to be changed. From this, e see some indication that a syntactic approach may play some role in aiding the adaptability of definition and that a semantic approach cold be more restrictive. From this research, e have seen ho the adaptability of a stdents concept image allos him or her to compare seemingly disparate concepts in ne contets. ere, the phrase seemingly disparate reflects the nderstanding of the concepts from the stdents initial points of vie. his action reflects inners interplay beteen definition and image, bt is different in that the participants ere not comparing a definition and image of a single mathematical concept, bt rather to different bt related images 1991, p. 70. his interplay is not addressed in his or, bt yields a reslt similar to that of inners interplay here an adaptation of image allos one to mae sense of a perceived relationship. n this case, the adaptation of to images alloed a relationship to be perceived. Conversely, in ettys case, rigidity restricted her perception of a relationship beteen independence and mtal eclsivity.

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