# Should I correct only one mistake at a time?

I am working as a teaching assistant in a course that first year mathematics students take. Some students who have maths as a minor also take it.

I run what is called a supervising session in the official translation - guidance session would be more literal translation. I help students who try to solve exercises similar to the ones they are to complete for the exercise sessions. There is no effect on grading or any other incentive besides learning for students to come to the guidance sessions. There are maybe 10 to 20 students (in maybe 6 to 13 groups, though some groups are singletons), and one tutor. I will visit the student groups, moving through them in order. Sometimes I will write a sample proof or discuss something several groups have problems with on the blackboard.

Often the students have several problems. Consider the following, which is very close to a real example, though most attempts are not quite this weak: The task is to prove that if $a \geq 4$ and $b \geq 2$, then $a+b \geq 6$. The student has written: $a \geq 4$, $b \geq 2$, $a+b \geq 6$. Antithesis: $a < 4$. After this the student has done calculations, some of them faulty, to get a contradiction with $a+b \geq 6$. The calculations are simply lines of inequalities without words or implication arrows joining them.

Some of the mistakes: Assumptions and the claim are not clearly separated. The antithesis is wrong. There are mistakes in the calculation. The writing is poor, as it only consists of formulae.

When encountering a student's proof attempt (or calculation) with several mistakes, should I focus on correcting one mistake at a time, or should I point out all of them?

I fear that a barrage of corrections will only confuse the students. On the other hand, correcting only something may not be enough for them to complete an exercise, and I may not guess what issue or which issues are worst ones.

Is there any literature on this, or any best practices?

I believe your intuition "that a barrage of corrections will only confuse the students” is on point. Even if you "may not guess what issue or which issues are worst ones”, you (who have a big-picture view of the ideas, how they relate, and where they are going) probably will have a fighting chance of diagnosing the issue (especially if you actively keep these things in mind and try to make sense of their thinking).

While "correcting only something may not be enough for them to complete an exercise”, that is not really the goal; the exercise, their response, and your corrections or feedback are tools to get to improved understanding on their part. A good reference on this is Dylan Wiliam’s Embedded Formative Assessment (http://www.solution-tree.com/embedded-formative-assessment.html); one of the main points he makes is that feedback should be more work for the recipient than the donor. New simpler questions are a great example of putting this into practice. Comparing responses — to a similar question, or better yet the same question they have already worked -- is another way, that helps give the learner a “nose for quality.”

I strongly believe that the best tutoring skill to have is to identify one issue they need to fix, then on the spot, craft a new, simpler question.

If you give them the right simplified question, they will make one error -- and as you point out, it is easy to correct one error. Once they feel confident, you can give them a new question in which they will make a new error. Then you can help fix that.

The hard part, of course, is being able to make up questions on the spot that are targeted at the student's exact problem.