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Relationships Between Mathematical Proof and Definition

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Relationships Between Mathematical Proof and Definition RELATIONSHIPS BETWEEN MATHEMATICAL PROOF AND DEFINITION 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 Probability 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 evidence 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 mathematics 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 definitions 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. Data 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 probabilities 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. Data Analysis 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 logic 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 integers 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.
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