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P. R. Halmos recalled a conversation with R. L. Moore where Moore quoted a Chinese proverb. That proverb provides a summation of the justification of the methods employed in teaching students to do mathematics with a modified Moore method (MMM). It states, "I see, I forget; I hear, I remember; I do, I understand." In this paper we build upon the suggestions made in, "On the Nature of Mathematical Thought and Inquiry: A Prelusive Suggestion" (2004, ERIC Document ED502336) and attempt to explore why the differences between reading, seeing, hearing, witnessing, and doing give rise to the contrast between and betwixt feeling, believing, interpreting, opining, and knowing. We refine in this paper the philosophical position proposed in the 2004 paper and accentuate how reading, seeing, or hearing do not lead to understanding whilst feeling or believing do not lead to truth. We submit that "interpreting" gives the impression Math is as imprecise as Psychology and is rooted in relativism (the "eye of the beholder") rather than certain conditional truth deduced from axioms. We posit that feeling, believing, & "interpreting mathematical phenomena," are actually harmful to authentic meaningful mathematical learning.

The author of this paper suggests several neoteric, unconventional, idiosyncratic, or unique approaches to beginning Set Theory that he found seems to work well in building students' introductory understanding of the Foundations of Mathematics. This paper offers some ideas on how the author uses certain "unconventional" definitions and "standards" to get students to understand the essentials of basic set theory. These approaches continue through the canon and are employed in subsequent courses such as Linear Algebra, Probability and Statistics, Real Analysis, Point-Set Topology, etc. The author of this paper submits that a mathematics student needs to learn how to conjecture, to hypothesise, "to make mistakes," and to prove or disprove said ideas; so, the paper's thesis is learning requires "doing" and the point of any mathematics course is to get students to do proofs, produce examples, offer counterarguments, and create counterexamples. We propose a quintessentially inquiry-based learning (IBL) pedagogical approach to mathematics education that centres on exploration, discovery, conjecture, hypothesis, thesis, and synthesis which yields positive results--students doing proofs, counterexamples, examples, and counter-arguments. Moreover, these methods seem to assist in getting students to be willing to make mistakes for, we argue, that we learn from making mistakes not from always being correct! We use a modified Moore method (MMM, or M[superscript 3]) to teach students how to do, critique, or analyse proofs, counterexamples, examples, or counter-arguments. We have found that the neoteric definitions and frame-works described herein seem to encourage students to try, aid students' transition to advanced work, assists in forging long-term undergraduate research, and inspirits students to do rather than witness mathematics. We submit evidence to suggest that such teaching methodology produces authentically more adept students, more confident students, and students who are better at adapting to new ideas. (Contains 36 footnotes.)

The author of this paper argues "is mathematics indispensable and are pre-requisites needed in mathematics courses?" that the answer to one question is, "yes," and to the other, "no;" however, the response to the questions is inverted in this sentence. Such responses might surprise some people; but, the author shall argue that learning requires doing; only through inquiry is learning achieved, and mathematical thought is one that must be focused on the process of deriving a proof, constructing an adequate model of some occurrence, or providing connection between and betwixt the two. The questions were posed in two provocative papers, "Do We Need Pre-requisites?" (1997) and "Is Mathematics Necessary?" (1997), which the author has read (seemingly) every year and has provoked much reflection. The answers to the questions will lead to a discussion of the Committee on the Undergraduate Programme (CUPM) 2004 Recommendations (2004). The paper is organised in the following manner. In the first part of the paper the author reviews the literature and offers a synopsis of the major points that are addressed in the paper. In the second part the author argues the main points: we need pre-requisites in mathematics courses; but, mathematics is not a necessary area of academia for all students. In the third part of the paper the author discusses the CUPM 2004 Recommendations and their relationship to the questions. Finally, the author proposes questions he believes should be addressed before the next set of CUPM Standards are produced. (Contains 2 figures and 94 footnotes.)

A course in the Calculus sequence is arguably the most difficult course in which inquiry-based learning (IBL) can be achieved with any degree of success within the curriculum in part due to: (1) the plethora of majors taking Calculus to which the sequence relates to their majors in what is considered an "applied" manner; and (2) the sequence is intertwined such that "coverage" matters since if a critical concept or area was not "covered" in Calculus I or II it might do serious harm to the student in Calculus II, III, or beyond where the understanding the topic may depend significantly on said material which was not "covered." So, this paper argues (pedagogical and practical justification are submitted) for use of a modified Moore method (MMM) which employs elements of the classic Moore method (students doing rather than seeing, hearing, or reading) which creates a moderate pace for the course; not too fast (as perhaps in a traditional German seminar (recitation) method) nor too slow (as perhaps in a constructivist or pure Moore method course) and presents the model for use the MMM in the Calculus sequence. Further, it is proposed that the MMM assists students to establish a firm foundation for subsequent course work and creates an excellent potential for students to have the possibility to master the material. The author of this paper has experienced teaching such courses in the Calculus sequence in such a manner for approximately twenty-five years. (Contains 45 footnotes.)

The author of this paper posits that inquiry-based learning (IBL) enacted via a modified Moore method (MMM) is a content-driven pedagogy; as such it is content-centred not instructor-centred or student-centred. The MMM is a philosophy of education where student must master material by doing; not simply discussing, reading, or seeing it and that authentic mathematical inquiry relies on inquiry though constructive scepticism. We must commit to conjecture and prove or disprove said conjecture; we must do in order to learn; through real inquiry learning is achieved, and hence this paper proposes an archetype of mathematical pedagogy such that the experience of doing a mathematical argument is the reason enough for an exercise and inquiry-based learning (IBL) enacted via a modified Moore method (MMM) is an authentic way to actualise a learning environment where the content studied is the centre of the experience. The pedagogy of IBL is like no other pedagogy (for others focus on manner of exposition, recitation, activities, exercises, etc. and less with the content as oft content is secondary to the other method(s)) because in IBL content is primary. Many methods of instruction are not active but rather passive and some students wish to be passive and do the least (work) for the most (highest grade). IBL cannot be done passively. For a student to master material it is necessary for the instructor to be a master of the material so that the instructor may guide students through the content; hence, the IBL is in the tradition of a master-apprentice system. The major focus of the paper is on how the use of the MMM creates a more effective mathematical education for students; how use of the MMM established an atmosphere that created for many students firm and authentic understanding of many of the principles of mathematics; and, therefore is key in mathematics education reform. We will also discuss how inquiry-based learning (IBL) enacted via a modified Moore method (MMM) is rooted in a philosophical position of "positive scepticism." What binds and supports mathematics is a search for truth, a search for what works, and a search for what is applicable within the constraints of the demand for justification. It is not the ends, but the means which matter the most - - the process at deriving an answer, the progression to the application, and the method of generalisation. These procedures demand more than mere speculative ideas; they demand reasoned and sanguine justification. Furthermore, "positive scepticism" (or the principle of "epoikodomitikos skeptikistisis)" is meant to mean there is a demand for objectivity; an insistence on viewing a topic with a healthy dose of doubt; a requirement for remaining open to being wrong; and, a stipulation that an argument may not be constructed or built from an "a priori" perception. Hence, the nature of the process of the inquiry that justification must be supplied, analysed, and critiqued is the essence of the nature of mathematical enterprise: knowledge and inquiry are inseparable and as such must be actively pursued, refined, and engaged. So, this paper proposes a pedagogical position that deviates from the "norm" insofar as it argues for inquiry-based learning (IBL) where the content studied is the centre of attention - - the student and the instructor should be secondary to the material in a university classroom where the experience of doing rather than witnessing is primary as it the case in an IBL-taught class. (Contains 38 footnotes.)

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 the complex plane as consequents of set theory. The connections between logic, set theory, and proofs about probability, random variables and processes, & 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 approximately 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 statistics 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 materials 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 education 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). (Contains 50 footnotes.)

The author of this paper submits that a mathematics student needs to learn to conjecture and prove or disprove said conjecture. Ergo, the purpose of the paper is to submit 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 (MMM) across the mathematics curriculum. The author of this paper has used the MMM in classes including an Introduction to Mathematics (general education liberal arts mathematics required as the minimum class that fulfils the mathematics requirement at Kutztown University of Pennsylvania (KUP)), Fundamentals of Mathematics I & II courses (mathematics for elementary education majors), Calculus I, II, \& III, Set Theory, Linear Algebra, Bridge to Higher Mathematics, Probability and Statistics I & II, Real Analysis I \& II, Topology, Senior Seminar, and Directed Reading. The author of this paper has taught for approximately twenty-five years, much of it at Morehouse College (MC) an historically black liberal arts institution, but now teaches at a comprehensive university in the Pennsylvania State System of Higher Education (PASSHE) where use of the MMM has been met with mixed reception by students and faculty. This paper discusses the techniques used to facilitate learning and the successes or lack thereof of how the methods and materials in the courses taught established a meaningful inquiry-based learning environment, how the method assisted in forging some long-term undergraduate research, and encouraged some undergraduates to delve into research who might not have otherwise embarked on research. So, this paper proposes an approach to mathematics education that centres on exploration, discovery, conjecture, hypothesis, thesis, and synthesis such that the experience of doing a mathematical argument, creating a mathematical model, or synthesising ideas is reason enough for the exercise - - and the joy of mathematics is something that needs to be instilled and encouraged in students by having them do proofs, counterexamples, examples, (informal) arguments, and counter-arguments in any mathematics course. Thus, the MMM used by the author is wholly a derivative of the Moore method and exists because of R. L. Moore, W. H. Mahavier, B. Fitzpatrick, M. Smith, C. Reed, D. Doyle, and other distinguished academics who instructed the author or the author's professors.

The author of this paper submits that a mathematics student needs to learn to conjecture and prove or disprove said conjecture. Ergo, the purpose of the paper is to submit 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 Bridge to Higher Mathematics course to teach students how 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 of the course with logic and predicate calculus as antecedents to the set theory, and number theory, cardinal and ordinal theory, or beginning topology of the reals as consequents of set theory. The author of this paper has experienced teaching such a course for approximately fifteen years; mostly teaching the course at a historically black college. The paper is organised such that in the first part of the paper justification for use of a modified Moore approach--both pedagogical and practical justification are submitted. In the second part of the paper the author submits the model for the Bridge course and focuses on what is effective for the students, what seems not useful to the students, and why; hence, explaining what practices were refined retained, modified, or deleted over the fifteen years. In the third part of the paper explanation is presented as to why the course was designed the way it was (content), how the course was revised or altered over the years and how it worked or did not for the faculty and students. The final part of the paper discusses the successes and lack thereof of how the methods and materials in the Bridge course established an atmosphere that created for some students an easier transition to advanced mathematics classes, assisted in forging a long-term undergraduate research component in the major, and encouraged some faculty to direct undergraduates in meaningful mathematics research. Qualitative and quantitative data are included to support what were or were not successes. So, this paper proposes a pedagogical approach to mathematics education that centres on exploration, discovery, conjecture, hypothesis, thesis, and synthesis such that the experience of doing a mathematical argument, creating a mathematical model, or synthesising ideas is reason enough for the exercise--and the joy of mathematics is something that needs to be instilled and encouraged in students by having them do proofs, counterexamples, examples, and counter-arguments in a Bridge course to prepare the student for work in advanced mathematics. (Contains 43 footnotes.)

This study investigated the predictive validity of the Descriptive Tests of Mathematical Skills (DTMS) and the SAT-Mathematics (SAT-M) tests as placement tools for entering students in a small, liberal arts, historically black institution (HBI) using regression analysis. The placement schema is four-tiered: for a remedial algebra course, college algebra course, pre-calculus course, and a first calculus course. Low psychometric indices were found between score earned on the Elementary Algebra Skills (EAS), Intermediate Algebra Skills (IAS), and Calculus Readiness: Functions and Graphs (CR) of the DTMS and grade earned in College Algebra, Pre-calculus, and Calculus I, respectively, and between score earned on the SAT-M and grade earned in College Algebra, Pre-calculus, and Calculus I. Define E denote the score earned on the EAS, I the score earned on the IAS, R the score earned on the CR, S the score earned on the SAT-M, B the grade earned in the Basic Mathematics course, A the grade earned in the College Algebra course, P the grade earned in the Pre-calculus class, and C the grade earned in the Calculus I class. The models for the DTMS produced were as follows: A{hat} = 1.504 + 0.039E for the EAS of the DTMS; P{hat} = 1.030 + 0.084I for the IAS of the DTMS; and, C{hat} = 1.670 +0.050R for the CR of the DTMS; whereas, the models for the SAT-M produced were: A{hat} = 0.241 + 0.004S; P{hat} = 0.695 + 0.003S; C{hat}= -0.519 + 0.005S. Perhaps the most interesting component of the study was not the regression equations, correlations, or any other parametric statistical portion of the study but the effect of the change of policy on the placement schema at the college. If placement decision were based on scores of SAT-M rather than on scores earned on the DTMS, there seems to be a shift of students such that less were placed in the remedial class and more into the courses that allowed a student to earn college credit and satisfy core curriculum requirements for mathematics. The paper ends with recommendations for further research on minimum competence testing and placement and notes the follow-up studies to this study that will be done by the researchers. (Contains 8 footnotes, 1 figure, and 5 tables.)