Table of Contents,Abstract ii, I Introduction Background and a Brief Overview of the Moore Method 1. II The Use of a Modified Moore Method in Probability and Mathematical Statistics Courses 6. III Organisation of the Courses When Using a Modified Moore Method 15. IV Summary and Conclusion 24,V References 28, A Modified Moore Approach to Teaching Mathematical. Statistics An Inquiry Based Learning Technique to Teaching. Mathematical Statistics,M Padraig M M McLoughlin,Department of Mathematics. Kutztown University of Pennsylvania, The author of this paper submits the thesis that learning requires doing only. through inquiry is learning achieved and hence this paper proposes a programme. of use of a modified Moore method in a Probability and Mathematical Statistics. PAMS course sequence to teach students PAMS Furthermore the author of this. paper opines that set theory should be the core of the course s pre requisite with. logic and calculus as antecedents to the set theory an introduction to the theory. of functions as subsets of R R as consequents of set theory The connections. between logic set theory and proofs about probability random variables and pro. cesses inferential mathematical statistics cannot be understated the better. the student s pre requisite knowledge the easier it is for the student to understand. probability theory and flourish in a Probability Statistics course sequence. The author of this paper has experienced teaching such a course sequence for ap. proximately fifteen years mostly teaching the course at a historically black college. The paper is organised such that in the first part of the paper an explanation as to. why Logic Set Theory and Calculus are proper pre requisites to a Probability. Statistics course sequence and a brief overview is presented of the Moore method. The second part of the paper presents justification for use of a modified Moore. approach in teaching probability statistics or what is termed mathematical sta. tistics often both pedagogical and practical justification is submitted In the third. part of the paper the author submits the model for the Probability Statistics. courses and focuses on what is effective for the students what seems not useful. to the students and why Also explanation is presented as to why the courses. were designed the way they were content how the courses were revised or altered. over the years hence explaining what practices were refined retained modified or. deleted and how such was helpful or not for the faculty and students The final part. of the paper discusses the successes and lack thereof how the methods and materi. als in the PAMS courses established an atmosphere that created for some students. an easier transition to graduate school preparation for actuarial tests to the work. force in applied statistics assisted in forging a long term undergraduate research. component in the major and encouraged some faculty to direct undergraduates in. meaningful research, So this paper proposes a pedagogical approach to mathematical statistics ed. ucation that centres on exploration discovery conjecture hypothesis thesis and. synthesis such that the experience of doing a mathematical argument creating a. statistical model or synthesising ideas is reason enough for the exercise and. the joy of mathematics and statistics is something that needs to be instilled and. encouraged in students by having them do proofs counterexamples examples and. counter arguments in a Probability and Mathematical Statistics course indeed in. any course, I Introduction Background and a Brief Overview of the. Moore Method, Mathematics is formally a branch of Philosophy under the Epistemology sub. heading As such a principle role in any mathematical education focuses the stu. dent on the attempt to deduce that which is conditionally true based on certain. assumptions axioms Whilst some mathematicians are concerned with Ontology. or Axiology namely is there an ultimate Truth to mathematics and can we dis. cover it what is the essence of the beauty of a proof its elegance brevity etc or. what is the value of the idea this paper presents the position that restricts the. discussion to the epistemological level as far as to deduce that which is condition. ally true present evidence to suggest the possibility that a particular pedagogy has. merit and argues that Probability and Mathematical Statistics should be studied. in a pseudo Socratic manner rather than studied as a Sophist would study a subject. Mathematics and in particular Probability and Mathematical Statistics PAMS. is built upon a foundation which includes axiomatics intuitionism formalism logic. application and principles Proof is pivotal to mathematics as reasoning whether. it be applied computational statistical or theoretical mathematics The many. branches of mathematics are not mutually exclusive Oft times applied projects. raise questions that form the basis for theory and result in a need for proof Other. times theory develops and later applications are formed or discovered for the the. Probability and Statistics mathematical statistics being a branch of mathe. matics therefore required an axiom system and thanks to Kolmogorov such was. produced From the point of the establishment of the basic axioms of probability. along with fundamental Aristotelian logic the Zermelo Frankel Cantor axioms of. Set Theory and the axioms of Analysis the theory of mathematical statistics be. came progressively more rigourous such that when we consider probability theory. we are able to say for example given S is a well defined sample space and E an. event it must necessarily be the case that the probability of the complement of E is. one minus the probability of E to use a facile corollary to the axioms of probability. theory as an example 1, Mathematical statistics education should be centred on encouraging a student. to think for one s self to conjecture to analyze to argue to critique to prove or. disprove and to know when an argument is valid or invalid Perhaps the unique. component of mathematics which sets it apart from other disciplines in the academy. is proof the demand for succinct argument from a logical foundation for the ve. racity of a claim Mathematical statistics is deductive science as is mathematics in. general whereas it is the case that Applied Statistics like its sister sciences Physics. Chemistry Biology Economics Psychology etc is fundamentally an experimental. science But Applied Statistics is founded upon the establishment of mathematical. truth specifically probability theory, The author of this paper submits that in order for students to learn students. must be active in learning Thus the student must learn to conjecture and prove. or disprove said conjecture One cannot learn to conjecture from a book we learn. 1S is a well defined sample space E an event P r E C 1 P r E. to conjecture by conjecturing 2 Ergo the author of this paper submits the thesis. that learning requires doing only through inquiry is learning achieved and hence. this paper proposes a philosophy such that the experience of creating an idea and a. mathematical argument to support or deny the idea is a core reason for an exercise. and should be advanced above the goal of generating a polished result. This paper outlines a programme of use of a modified Moore method MMM. in a na ve Probability Statistics P S course sequence3 4 to teach students. about axiomatic probability conditional and marginal probability mutual exclu. sivity statistical independence random variables discrete and continuous mul. tivariate distributions moments probability mass functions PMFs probability. density functions PDFs moment generating functions MGFs cumulative dis. tribution functions CDFs conditional and marginal PMFs or PDFs etc by. teaching them to do critique or analyse proofs counterexamples examples or. counter arguments Furthermore the author of this paper opines that Set Theory. should be the core antecedent to the course with logic and Calculus as the other. antecedents to the P S courses and ideally a basic understanding of functions. which are a subset of R R to compleat a foundation on which rises the Probability. Statistics, R L Moore created or adapted a pseudo Socratic method which bears his name. 13 18 19 21 31 60 61 and 62 5 He said that student is taught the. best who is told the least 6 It is the foundation of his philosophy and it sums up. his philosophy of education simply tersely and succinctly Moore believed that the. individual teaches himself and the teacher is merely an informed guide who must. not trample on the individual s natural curiosity and abilities 7 The Moore method. accentuates the individual and focuses on competition between students Moore. himself was highly competitive and felt that the competition among the students. was a healthy motivator the competition amongst students rarely depreciated into. a negative motivator and most often it formed an esprit d corps where the stu. dents vie for primacy in the class 8, This statement is not meant to be sarcastic but to illustrate how fundamental to the argument. forwarded in this paper that the act of conjecturing is central to inquiry based learning IBL. Also oft titled Mathematical Statistics I and II Theory of Probability Statistical Theory or. some other title We assume it is a junior senior or first year graduate course sequence that we. are discussing herein that is designed to delve deeply into the underpinnings of Applied Statistics. We designate a na ve Probability Statistics course or courses to mean that Lesbegue measure. theory and probability measures are not pre requisite or discussed within the courses. 5Some have argued that his advisor E H Moore no relation might be rightly credited with. inventing the Moore method This is of no interest to the author it suffices that such a method. exists for the sake of this paper, 6 R L Moore Challenge in the Classroom Providence RI American Mathematical Society. 1966 videocassette See also Miriam S Davis Creative Mathematics Instruction Ed D dis. sertation Auburn AL Auburn University 1970 25 Benjamin Fitzpatrick Jr The Teaching. Methods of R L Moore Higher Mathematics 1 1985 45 and Lucille S Whyburn Student. Oriented Teaching The Moore Method American Mathematical Monthly 77 4 1970 354. 7 See Davis Creative Mathematics Instruction and Paul R Halmos How To Teach In I Want. To Be A Mathematician New York Springer Verlag 1985 for a more detailed discussion of. Moore s tenets, 8See Davis pages 21 and 119 and D R Forbes The Texas System R L Moore s Original. Edition Ph D dissertation Madison WI University of Wisconsin 1971 pages 168 169 172. and 188 for a detailed psychological analysis of the formation of esprit d corps from competition. Many educators opine that the Moore method is perhaps best suited for graduate. level work where there is a rather homogenous set of students who are mature. Moore s philosophy of education is too often considered a method of teaching and. as such can be adopted and practiced The author opines that this is an error and. it is a philosophy of education Therefore adoption of the methods that Moore cre. ated and practiced would be meaningless and could lead to harm for the students. if the practitioner did not subscribe to Moore s philosophy Whyburn rather poet. ically notes that Moore s beliefs gives one the feeling that mathematics is more. than just a way to make a living it is a way of life an orderly fashion in which you. want to consider all things 9, Given the harshness of the Moore method or due to the change in American. education since Moore s death 1974 there have been several proposed modified. Moore methods MMM which are similar to but not identical to the one proposed. herein for teaching mathematical sciences courses 1 2 17 and 20 which could. be used in a mathematics computer science physics or statistics course If one. agrees with the philosophical position conditional to the modified Moore method. then it is an entirely acceptable teaching methodology However it still requires. that the individual learn without the aid of books collaboration subject lectures. and demands uncompromisingly talent from the individual 10. The use of books would cause the student to be a witness to mathematics rather. than a participant 11 Since the Moore method is based on the assumption that this. talent is dormant or latent within the student 12 the student is expected to do all. that is necessary to tap into this dormant talent There is not an expectation on the. part of Moore s philosophy that this dormant talent will awaken easily or quickly. Thus the pace of learning is set by the student or students 13. Several authors opine that a constructivist approach to teaching mathematics or. statistics courses 24 47 53 54 and 56 is the proper method The construc. tivist accentuates the community and focuses on cooperation amongst students. The constructivist approach includes alternate assessment group projects service. learning etc and closely resembles pedagogically the National Council of Teachers. of Mathematics 44 standards and Dewey s position 14 15 and 16 If one. agrees with the philosophical position conditional to the constructivist method. then it may be an entirely acceptable teaching methodology It seems that the. constructivist method is best suited for elementary or s. A Modi ed Moore Approach to Teaching Mathematical Statistics An Inquiry Based Learning Technique to Teaching Mathematical Statistics 479 300697 M Padraig M M McLoughlin Ph D 265 Lytle Hall Department of Mathematics Kutztown University of Pennsylvania Kutztown Pennsylvania 19530 mcloughl kutztown edu Paper presented at the Annual Meeting of the American Statistical Association

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