Keith Devlin of Stanford University defines mathematical proof this way:
Proofs are stories that convince suitably qualified others that a certain statement is true.
This quote is from a blog post entitled “What is a proof, really?“, and I find it to be a useful definition. In this class, you may find it helpful to consider your classmates to be the “suitably qualified others.”
Here is a rubric for evaluating proofs, adapted from Keith Devlin’s:
| Novice | Apprentice | Practitioner | |
| Logical correctness | Solution is fundamentally flawed. | Solution is mostly fine, with one or two slips of logic. | Solution is complete and correct. |
| Clarity | The argument is difficult to follow. There may be dangling sentence fragments, or a sequence of equations with no clear flow. | The argument can be found, but only with some work on the part of the reader. | Solution is clear and easy to follow. The writer uses complete sentences throughout (“=” is a verb); sentences start with words (not symbols). |
| Opening /Closing | The reader has to infer what’s being proved and when the argument is done. | There is an opening and a closing statement, but they are muddled and/or misplaced. | There is a clear statement at the beginning of exactly what is being proved, and a clear statement at the end that the result has been established. |
| Reasons | Reasons for statements are not provided. | Some reasons are provided, but there are big jumps and/or unnecessarily detailed justifications. | Reasons are given for all significant steps. |
| Efficiency | The argument is repetitive or includes unnecessary information (which may be true, but is not helpful). | The progression is mostly smooth. There may be one or two extraneous statements. | The solution is elegant: complete while free of any distractions; the readers says “this makes it seem simple!” |
You can use this to write and revise your own work, and to give responses to peers when I ask you to do that. Keep in mind that improvement will take time and practice!
Selected Solutions
Here are the solutions to the first exam: Math302SpringExam1Solutions.
Here’s a proof that our definition of even and odd permutations makes sense: Even/Odd Permutations.