Mathematical problem solving is very much like a journey. One is presented with a problem and must work out a solution to reach the other side.
Using this analogy, this 4-point rubric is a visual way for students to understand their progress.
I developed this system in 1997 when teaching 4th grade. At that time I was incorporating math and writing using manipulatives and open-ended questions. With the return of this style of teaching under Common Core, I have dusted The Journey Rubric off and revised it for today's math teacher and students.
It is important with any rubric or grading system to fully instruct your students in what you deem is important for them to show they have reached mathematical understanding of the task involved. (Rubric is intentionally open-ended to be used over and over for all types of problem solving activities. You are free to revise wording to make graphics problem specific. However, students are often quite divergent and will solve the problems in ways you would never expect. That is why I prefer a clear explanation of process and show the arithmetic.)
When using this rubric, I rarely ever accept papers that score a 1 or 2. I reteach concept, have them walk me through their process. I will help facilitate their progress.
When students try to turn in a 3 paper, student will walk me through their process, and if the spoken explanation does not match the written response I will send students back to desk to polish up their paper. Or, if the explanations are well written, but student made a careless arithmetic error, I send them back to check their algorithm. They are always happy to do this as they know these tiny changes mean a 4 paper.
When a student turns in a 4 paper, they often skip across the classroom to their desks singing, "I made the journey!"
(I accept a 1 or 2 paper only when we have run out of class time. Then, I know which students need remedial attention.)