By the end of Transition to Higher Mathematics, students should be able to write proofs and grasp the role of proof as a formal mathematical explanation.
The course builds toward having students prove more logically complex statements and gaining facility with different kinds of proof. The recurring big ideas surround how to attack a proof. We go into depth on key tools like using examples, applying the forward-backward method, using the logical structure of the statement, using definitions.
All of the chapters support these main goals. Students are first introduced to the idea of proof through familiar ideas of number theory and divisibility. They then gain some initial background in logic, and apply it to some number theory proofs before moving on to sets. They are then asked to apply set ideas to sets of real numbers. Then they move to the critical mathematical idea of a function and write proofs about functions and their properties. Finally they are briefly introduced to equivalence relations before moving on to looking at other techniques of proof, and applying these ideas to concepts already seen in class.
Given my priority goals, assessments need to determine whether students are able to demonstrate an understanding of the key approaches to proof, the structure of a logical argument, and to explain the key mathematical concepts of set, open set, function, and properties of a function.
Given my goals, I have exercises and follow-up tasks that should help students gain insight into how the main ideas in key proofs are put together.
If I have been successful, students will be able to transfer their learning to upper division mathematics courses that follow, by attacking proofs with confidence and awareness of the tools available to them, and will persist through difficult courses in the major. If I have been successful, students should avoid such common issues as waiting for the professor to tell them what to do, believing that only others can write proofs, and being uncertain of the role of examples in generating proof ideas (vs. using examples as proof).
I enjoyed completing this exercise for this course. The course is built as a coherent whole, telling the story of proof and its role in mathematics. To make this work, I have done some tinkering with the emphasis areas of the course, reducing time spent on truth tables (which have a role, but it need not be multiple weeks of a course), and trying not to spend too much time proving things that are too basic.
What about your courses? Does anyone want to take this up with one of their math courses?