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Teaching Discrete Mathematics Entirely from Primary Historical Sources

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Teaching Discrete Mathematics Entirely from Primary Historical Sources DISCRETE MATHEMATICS 1 TEACHING DISCRETE MATHEMATICS ENTIRELY FROM PRIMARY HISTORICAL SOURCES Janet Heine Barnett Mathematics and Physics Colorado State University { Pueblo Pueblo, CO 81001, USA [email protected] Guram Bezhanishvili Mathematics New Mexico State University Las Cruces, NM 88003, USA [email protected] Jerry Lodder Mathematics New Mexico State University Las Cruces, NM 88003, USA [email protected] David Pengelley Mathematics New Mexico State University Las Cruces, NM 88003, USA [email protected] 2 J. Barnett, G. Bezhanishvili, J. Lodder and D. Pengelley TEACHING DISCRETE MATHEMATICS ENTIRELY FROM PRIMARY HISTORICAL SOURCES Abstract: We describe teaching an introductory discrete mathematics course entirely from student projects based on primary historical sources. We present case studies of four projects that cover the content of a one-semester course, and mention various other courses that we have taught with primary source projects. Keywords: primary historical sources, original historical sources, peda- gogy, discrete mathematics 1 INTRODUCTION In [7] we described teaching undergraduate mathematics using student projects based on primary historical sources. We presented our moti- vation, our pedagogy, the advantages and challenges of teaching with primary historical sources, student responses, and three sample project case studies. All our projects and how we have used them in various courses may be found in [2, 3, 4, 5, 23]. Our student projects are modular, and can be used to replace even a small part of the textbook material in a course. Among the myriad positive student comments about learning mathematics from primary sources, remarks like the following inspired us to aim higher, for more than just small parts of a course: You don't learn to play an instrument without listening to those who played it best. 4 J. Barnett, G. Bezhanishvili, J. Lodder and D. Pengelley It gives you the sense of how math was formed which prepares you for how to think up new, innovative mathematics for the future. It puts the ideas in context where textbooks fail. Textbooks are like inventions without instruction manuals. In this sequel paper to [7], we describe and analyze how we went further, developing enough project material to eventually teach several standard one-semester courses entirely from our projects, jettisoning standard textbooks. The courses we have taught entirely from projects based on primary historical sources are lower division discrete mathe- matics and upper division combinatorics and mathematical logic. Here we will present four projects that we have used in multiple ways to teach an entire introductory discrete mathematics course. As we built modular projects into existing courses, and eventually to fill entire courses, we evolved a particular guided reading approach common to all our student projects. With our own writing we guide the student in their own reading and study of selected primary source excerpts. A key feature is that we interrupt the primary source with clarifying comments, and with exercises. The exercises often have an in- tentional didactic purpose, provoking students to gain through their own intellectual work specific critical insights and understanding from their guided engagement with key spots in the primary source. This enables better understanding of both the source and the mathematics for going forward. The exercises may address aspects of rigor and proof, reflection on the evolution of mathematical processes, of terminology, notation and definitions, and other components of developing mathematical compe- tency that can emerge from guided sense-making of a primary source. Our projects also have a number of other design features discussed at length in [6, 7, 8]. While all our guided reading projects have pedagogical commonali- ties as detailed above, they also have substantial individual differences, particularly in the choices made for how to incorporate primary source material. The reader may look for these as we present the four projects featured below. In the Conclusion we will discuss the nature of the DISCRETE MATHEMATICS 5 differences between these four projects. 2 FOUR CASE STUDY STUDENT PROJECTS IN DIS- CRETE MATHEMATICS WITH PRIMARY SOURCES We have taught an entire one-semester introductory discrete mathemat- ics course from primary source projects. Standard topics in such a course often include basic logic, sets, functions, relations, elementary number theory, and proof techniques with special emphasis on mathematical in- duction. The following four of our projects fit well with these topics: 1. \Deduction through the Ages: A History of Truth," with sources by Chrysippus, Boole, Frege, Russell and Whitehead, Wittgenstein, and Post. 2. \An Introduction to Symbolic Logic," with sources by Russell and Whitehead. 3. \An Introduction to Elementary Set Theory," with sources by Can- tor and Dedekind. 4. \Pascal's Treatise on the Arithmetical Triangle: Mathematical In- duction, Combinations, the Binomial Theorem and Fermat's Theo- rem." We have used various combinations of these projects to teach a 3- credit 15-week one-semester course. The course has three basic units: logic, set theory and functions, and mathematical induction. Each unit occupies roughly four to five weeks of the course. The logic unit consists of basic propositional logic and quantifica- tion. This is covered by either the first project and the second half of the second project, or the entirety of the second project. The unit on sets and functions is covered by the third project, and the unit on mathematical induction is the main thrust of the fourth project, which also includes optional number theory. We usually begin with the logic unit, but we have taught the course with the latter two projects in either order. Each project has detailed notes to the instructor, which outline different possible uses of the project in and out of the classroom. In 6 J. Barnett, G. Bezhanishvili, J. Lodder and D. Pengelley the subsequent four subsections we will present overviews, topical high- lights, and sample primary source excerpts and student exercises from these four projects. Each subsection title consists of the italicized title of one of the four projects, along with an indication of some of the key topics featured. We have also developed and used several additional projects in teach- ing discrete mathematics. In the Conclusion we will discuss how these may be combined to create alternative course syllabi. 2.1 Deduction through the Ages: A History of Truth: Rules of Deduction, Logical Implication, Propositional Logic and Truth Tables Almost every introductory discrete mathematics course covers elemen- tary propositional logic, including the truth table for an implication (an \if-then" statement), often written p ! q, read \p implies q." The truth values of an implication when the hypothesis p is false are counter- intuitive and often a stumbling block for students, not to mention in- structors, who struggle to justify these truth values with strained exam- ples of implications, suggesting that truth tables are sometimes verifiable by mere example, while at other times requiring rigorous computation. The status of a logical implication in today's textbooks is that of a defini- tion, although the intellectual struggle leading up to the common accep- tance of this definition is almost never discussed in modern textbooks, and certainly not from the history of deductive thought. While such a history could encompass many possible sources, the teaching project \Deduction Through the Ages: A History of Truth" [21] offers excerpts from a few selected sources from Antiquity to the twentieth century that illuminate and focus how deductive reasoning has been reduced to the truth values of one type of statement, the logical implication. The project opens with five rules of deduction stated verbally by the ancient Greek philosopher Chrysippus (ca. 280{206 b.c.e.), which for the reader's convenience are reproduced below [17, p. 189] [22, p. 212{213]. DISCRETE MATHEMATICS 7 11111111 1. If the first, [then] the second. The first. Therefore, the second. 2. If the first, [then] the second. Not the second. Therefore, not the first. 3. Not both the first and the second. The first. Therefore, not the second. 4. Either the first or the second. The first. Therefore, not the second. 5. Either the first or the second. Not the first. Therefore, the second. 11111111 The project offers examples of these rules, but more importantly, asks the student why five separate rules are necessary and which rules, if any, are equivalent. Insight into the possible equivalence of certain rules is gained by reading selected historical sources, and forms the leitmotif of the project. Verbal argument asserting the equivalence of Chrysippus's rules is difficult and a more symbolic method of discourse is sought. The next selection in the project is from George Boole's (1815{1864) opus An Investigation of the Laws of Thought [11, 12], where we witness an initial attempt to reduce language to a symbolic format. Boole bor- rows the symbols ×, +, − form algebra, roughly denoting \and," \or" and \not" respectively, although the symbols no longer maintain all of their algebraic proprieties, but satisfy \the laws of thought." With this source, progress is made in the reduction of verbal argument to calcula- tion, although the equivalence of Chrysippus's rules remains open. An entirely different system of notation, the “Begriffsschrift" or concept- script is introduced in the work of Gottlob Frege (1848{1925) [18, 19]. The center piece of his notation is the condition stroke, written ξ ζ ; 8 J. Barnett, G. Bezhanishvili, J. Lodder and D. Pengelley whose truth value depends on the truth values of the individual argu- ments ζ and ξ. The entire symbol is true, except when it begins with a true statement (ζ) and ends with a false statement (ξ). In this case only is the condition stroke false. This compares well with Philo of Megara's (ca. 4th century b.c.e.) verbal statement that a valid hypothetical proposition is \that which does not begin with a truth and end with a falsehood" [29, II.
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