Proof: Introduction to Higher Mathematics
Why Use This Text? This text includes the usual “transition to higher mathematics” topics. However, there are numerous reasons it is significantly different from other texts with the same goals.
1) discusses the entire language of mathematics and does not assume the students are already fluent.
2) emphasizes precisely the logic and logical equivalences that are commonly used in mathematics. It explicitly shows how statements are commonly rearranged for proofs. It uses the logical forms of theorems to indicate how to organize proofs, especially, where to begin proofs and where to end them.
3) uses many conjectures to involve students in deciding if statements are true or false and to force them to be critical (e.g., conjectures pose the converse of a theorem, or something similar with a hypothesis omitted, or something that looks likely but is actually false).
4) has many homework problems clearly marked (with ☺) as short-answer problems that are excellent for students to do aloud in class. They can be used to involve students and give immediate feedback.
5) makes the useful distinction between a “concept image” and a “concept definition,” where the image is informal and the definition is formal. Proofs use precise concept definitions.
6) clarifies how formal definitions work, with many simple examples that illustrate how definitions are used in proofs.
7) clarifies all the facets to “translation,” because two sentences can look different and say the same thing.
8) is clear about which prior results can be used in a proof by going deep enough into particular topics so students have a foundation to build on.
9) is clear, when writing proofs, about what part is the proof and which parts are instruction about how to do proofs. [Our instructional comments are distinguished by enclosing them within brackets.]
10) repeats good ideas and advice because students will not grasp everything the first time around.
Homework. ☺ The symbol “☺” marks problems that are very short and could be done aloud in class. Your students will rapidly find out if they are doing it right, and so will you.
* An asterisk marks problems that are conceptually important.
Supplements. Faculty who adopt this text will have access to many useful supplements, including an Instructor's Solutions Manual and student manuals in pdf form by chapter that may be provided gratis to the students, or not. We will "share" a dropbox folder with you. It has one semester's HTML used for communicating with students, with a full calendar and homework assignments, comments, handouts on proofs, numerous old exams, notes about exams, etc.
Problems are classified as A, B, or C problems. “A” problems should be easy. “B” problems reflect the current level of work and understanding expected of the students. “C” problems are not required. However, they are interesting and may be assigned if you wish to go deeper into the material and the students are capable.
Parts of Chapter 3 and all of Chapters 4 through 8 are designed to be somewhat like mathematics research. Many results and conjectures in those sections are unproven or unresolved (the resolutions are homework problems). Asking students to resolve them in class is an option.
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