It is confidence that allows students to make decisions in math, to decide how to tackle a problem, to believe they can tackle a problem. If they have no confidence, they will take no risks. To believe they can do math in the present or the future, they have to believe they have been able to do it in the past. In encouraging this belief, it is more useful to the student if the instructor is ready to notice and comment on what they have done right, rather than what they have done wrong. Part of that mindset is to cultivate the habit of cutting math problems into small pieces, so that the small parts they did correctly can shine out, and you can acknowledge them.
Students are aware of how you respond to other students. Responding positively to one student may prompt another to take a risk, anticipating that your response will be similar to the one he has witnessed.
Ideally, I would mark each question as the student does it. When students start to work on a page in their books, or on a sheet I have given them, I don’t wait for them to finish and hand it in, rather I move around immediately, marking each one as I go, then coming back when they have a few more questions done. When I find several people having the same difficulty, I can call the class together, apologize for not having been clear enough in my explanations, and go over it again.
When I am working one-on-one to mark a student’s work, my job is to be encouraging by pointing out what the student has done correctly, and what evidence I see of good math thinking. My job is also to give the student a chance to articulate what she is doing, to help her remember, and to give her control over the process.
When I look at a page with the student, first, I put a checkmark beside every correct answer. I ignore the ones that are wrong, and comment that the student got some/many correct. If there are only a few wrong, I start with the first one that is correct.
I ask the student to tell me how she did it, so she has a chance to “rehearse” the procedure and articulate it clearly. I repeat with the next correct one, and the next, until she can easily articulate the procedure. Then I move to the first one that was wrong, and again ask the student to explain what she did. Usually, she will find her mistake and correct it on her own. I mark it right, and praise her for being able to find her own errors without me pointing them out. I continue with the next one wrong; at some point, I can ask her to independently check all the ones that don’t have a checkmark beside them.
If more than a few questions are wrong, I find the first one that is right, mark it right, and ask the student if I can guess what she did. I go over the question, teaching and dialoguing with the student about the method. I try to figure out where the student’s error is coming from. I then present a new question to the student and ask her to do it while I watch/coach. Then another. When I am sure the student has the process in mind, I offer a clean copy of the worksheet to do, or a new worksheet with similar problems.