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.)