Commit a7536999 by Jim Hefferon

update draft sent to JIBLM

parent 84656877
 ... ... @@ -233,7 +233,7 @@ \tableofcontents \chapter*{Preface}\pagestyle{plain} \chapter*{Preface} % \pagestyle{plain} \addcontentsline{toc}{chapter}{Preface} This is a course in proof writing ... ... @@ -247,23 +247,23 @@ Students get a sequence of things to prove, along with definitions and a few remarks. They attempt these outside of class, and then in class they propose solutions as well as carefully examine the solutions proposed by others. The instructor only lightly guides and the students pledge not to use outside sources. solutions proposed by others (the instructor only lightly guides and the students pledge not to use outside sources). Together, they talk through misunderstandings, sometimes stumble in the dark, and sometimes have beautiful flashes of insight. In short, they \emph{do} the mathematics. The advantage is twofold. There are two advantages. First, students own it\Dash they are completely engaged. This is the ultimate in active learning. Second, through their discussions, students come not only to see what is right, but also students come not only to see what is right but also to understand why what's wrong is wrong. For these students, with this material, this is the best way to develop mathematical maturity. Besides, it is lots of fun. Besides, it is a lot of fun. \medskip ... ... @@ -273,7 +273,7 @@ We cover elementary number theory, sets, functions, and relations. We start with number theory instead of sets for the same reason that the baseball team's annual practice starts with tossing the ball and that the baseball team's practice starts with tossing the ball and not with reading the rule book. Math majors take readily to proving things about divisibility and primes, ... ... @@ -296,18 +296,18 @@ This level gradually rises. Some exercises have multiple items; these come in two types. If the items are labeled \textsc{A}, \textsc{B}, etc., then each one is hard enough to be a separate assignment and in class I ask a different student to put a proposed solution on the board for each item. and in class I ask a different student to propose a solution for each. If the labels are i, ii, etc., then I ask a single person to do them all. a single person does them all. \medskip \noindent\textbf{Acknowledgments.} The material is standard but I must recognize my debt to the wonderful presentations of J~Jensen-Vallin and D~Velleman. I am also glad to express my debt to the Saint Michael's students, who have taught me a great deal. I am also glad for the chance to thank the students of past classes in Proofs, who have taught me a great deal. ... ... @@ -336,30 +336,29 @@ students, who have taught me a great deal. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%To the Instructor%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{annotation} \chapter{For instructors}\pagestyle{plain} \chapter{For instructors} % \pagestyle{plain} % This text is for a class using Inquiry-Based Learning. I will describe here how I use this book, but many variations are possible. For a fuller exploration, visit the site of I will describe how I use this book, but many variations are possible. For a fuller exploration, visit the web site of \href{https://www.comathinquiry.org/}{Communities for Mathematics Inquiry in Teaching Network}. I cover the first three chapters. (It is a semester course, meeting three times a week.) The fourth chapter works as a lecture on the final day or two, or perhaps as an honors add-on. We run a semester course, meeting three times a week. I cover this book's first three chapters. (The fourth chapter works as a potential lecture on the final day or two or as an honors add-on.) At the start, At the first meeting, I explain that in this course we develop each person's ability, as a future professional, to work independently. Students pledge that they will not work together outside of class and will not use any resources such as other books or the Internet. Consequently, on each day, each person arrives having worked Consequently, each person arrives to each class having thought carefully about each exercise, on their own. Usually there are four exercises. I randomly pick students to put proposed arguments on the board. put proposed solutions on the board. (I shuffle index cards. The picked students negotiate among themselves over who will do which one, a student picked in the prior class will not be picked today, ... ... @@ -396,17 +395,17 @@ next class, to be graded. I will close with a few comments. One is that in the first two weeks, I start with fifteen minutes of working through I start class with fifteen minutes of working through slides that cover elementary logic (these classes usually have extra time). (the early classes usually have extra time). These provide a vocabulary and sharpen the discussions, particularly about fine points such as vacuous implication. The second comment is on who is in the class. We enroll sophomores who have had Calculus~III and often Linear Algebra, so we can expect that they have an aptitude as well as the basic scaffolding Linear Algebra, so we can expect that they have an aptitude, as well as some basic scaffolding of mathematical reasoning on which to base this class's development. We limit to twenty students because in a too-large class ... ... @@ -419,8 +418,8 @@ Finally, about grading. I count four things: in-class contributions, the weekly hand-in problems, an in-class midterm, and an in-class final. In all four, I give credit both for mathematical correctness and for rhetorical competence. I weigh the four equally for mathematical writing. I weigh the four equally, but students agree that the in-class discussion is the core experience. Success flows from that. ... ... @@ -440,10 +439,10 @@ That's what I do. The home page also has other material that you may find useful, including the elementary logic slides and a guide to the basics of writing Mathematics. a guide to the basics of writing mathematics. All of the materials come with \LaTeX{} source so that you can adapt it to your needs. them to your needs. I am always glad to get reports, either a description of your experience, or suggestions, or bugs. ... ... @@ -4192,7 +4191,7 @@ is one and only one associated domain member, is a Prove these for a function~$f$ with a finite domain~$D$. They imply that corresponding finite sets have the same size. sets have the same cardinality. \hint for each, you can do induction on either $|D|$ or $|\range(f)|$. \begin{exes} \begin{exercise} ... ...
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