Coach students to create solutions by asking them to explain answers or how they arrived at conclusions instead of accepting straightforward answers. When assigning a grade, you want to consider the presentation of the solutions in addition to the correctness. Much of mathematics is about the process of thinking rather than simply crunching numbers.
Students coming into any mathematics course usually have little confidence in their own abilities to explain mathematics. If you are too rigid in what you will accept, the result will be to squash the confidence that you are trying to build. When a student asks, “What did they want?” Answer by saying, “They wanted to find out what you wanted to say.” It works! When you are grading a problem that asks for a student explanation, be as liberal as you can in grading it .
Allow students to resubmit a homework especially when it is evident that they missed key concepts of the lesson with comments to lead them on the right path. Anecdotal evidence suggests that actual need for the resubmissions taper off over time. In fact, you can decide on a policy to only allow multiple attempts in the first two units.