5: All parts of the problem are present. The solution is complete or near-complete and logically formulated. Explanations are clear, notation is correct and consistent, and conceptual understanding is apparent. The solution is not necessarily perfect; minor write-up problems, grammar errors, or arithmetic errors may be present.
4: Almost all parts of the problem are present. The solution is complete or near-complete and logically formulated. Explanations are mainly clear, notation is mostly correct and consistent, and conceptual understanding is apparent. Careless mathematical errors may be present (for example, algebra, arithmetic). An intelligent, thoughtful, but somewhat incorrect approach may receive this score. The difference between a 4 and a 5 is that a 4-solution has one significant flaw. The difference between a 4 and a 3 is that a 4-solution exhibits better conceptual understanding.
3: The solution is logically formulated with clear explanations,
although one part of the problem may be missing. OR, The solution is complete
and explanations are clear but there are serious logical flaws. OR, The solution
may be complete and logically formulated but the explanations are unclear or
Conceptual understanding of essential ideas is adequate. Many careless mathematical errors may be present (for example, algebra, arithmetic). Details are confused or missing.
2: Explanation is lacking. OR, The solution shows serious
misconceptions and some correct reasoning. OR, Many parts of the problem are
Conceptual understanding is inadequate. Procedural errors are present, OR logical/relational steps are missing. The difference between a 2 and a 3 is that a 2-solution does not exhibit adequate conceptual understanding of the essential details. The difference between a 2 and a 1 is that a 2-solution has some explanation, or has some correct reasoning.
1: Explanation is completely missing. OR, Many parts of the
problem are not present. OR, The solution contains serious logical flaws, and
lacks some explanation.
Conceptual understanding is inadequate. Procedural errors are present, OR logical/relational steps are missing. Poor response to the questions posed.
0: Solution is missing. OR, Solution is minimal and makes no sense.
This is an adaptation of the famous document How Doug Marks Papers (which appears in print in The Michigan Calculus Program Instructor Training Matierials by Beverly Black, Pat Shure, Doug Shaw). Doug Shaw wrote an explanation of his grading practices for fellow instructors, as a favor to them. I've been using that document as a guideline, modified very slightly, for many years. However, it's not a document that would make a lot of sense to students. Thus, this is sort of a translation of his intent (or at least my attempt at a translation) for use by students. In using How Doug Marks Papers, one first comments/corrects the problem and then assigns a numerical grade. This document is intended to help students to understand what the numbers on their homework mean as an assessment by an instructor of the overall quality of the work. It is also informed by Emert and Parish's MAA Notes article, available in this PDF book.