Hello Joseph , 

erros are signs of hypothesis making and testing and can be used to bolster flexibility , creativity and  risk taking  , they may  function as inevitable part of any learning activity for teachers who act as facilitators.  

I would say that the problem is not how to use error analysis but that is so rarely used. For example, many Moodle courses offer quizes for students, to check their understanding of a topic. The results (errors?) are in most cases not used by teachers while preparing to teach the next topic (or other connected topics). So I think more work should be done on facilitating the usage of results (visualisation?, reports?).

Teachers should consider the constructivist view on errors as potentially creating productive opportunities to understand and tap students' resources.

Borovcnik, M., & Bentz, H.-J. (1991). Empirical research in understanding probability. In R. Kapadia & M. Borovcnik (Eds.), Chance encounters: Probability in education (pp. 73-105). Dordrecht, Holland: Kluwer.

Hammer, D., Van Zee, E. (2006). Seeing the science in children’s thinking: case studies of student inquiry in physical science. Portsmouth, NH: Heinemann. [Chapter 2]

Smith, J. P., diSessa, A. A., & Roschelle, J. (1993). Misconceptions reconceived: A constructivist analysis of knowledge in transition. Journal of the Learning Sciences, 3(2), 115-163.

This is much in sync with some of the earlier answers.

Errors are like symptoms of diseases.  They can be caused by more than one thing, but the number of causes is potentially small since we can eliminate lots of reasons why the error might have occurred and focus on a few.  For example there are a small subset of things that can cause a high fever.  However simply providing aspirin doesn't eliminate the underlying problem

So, with an error, as with a fever, if we provide remediation for the most likely cause and that fails then we can often move to the next likely cause and find a remedy for the cause of the problem.  However, simply providing correct answers may only relieve the symptom without eliminating the underlying problem. 

So I'd say the key thing about errors is recognizing that they can be a teacher's best friend and help point the way to learning.  Much of the rest is technique and personal style.

To build on what Thomas said, teachers can utilize formative assessments to inform the next step in instruction.  One way to encourage students who make errors in calculations is to adopt the practice of "my favorite no."  The teacher looks through student errors and finds the one (either the most common or the one which lends itself best to clearing up confusion, and presents it to the class as "my favorite no," pointing out what the student did well and then where their thinking took a wrong turn. (The student is not identified).  By utilizing mistakes in this manner, the entire class learns from the error.  They can look at their own work and examine where their thinking took them in approaching the problem. The other benefit of this practice is that teachers begin to see patterns in student responses and know how to change their instruction to anticipate common errors.  

As I understand the "error analysis" concept, it calls for teachers to have the flexibility to "interview" their students in response to the computational errors on their problem solving, in regards to all the basic skills in addition, subtraction, multiplication, and division. I read that some students have memory deficits that conflict with their ability to discern the actual solution from their error maligned response. Happy new Year!

Raffaella Borasi has written about using errors as springboards for learning mathematics; see:

Borasi, R. (1996). Reconceiving mathematics instruction: A focus on errors. Greenwood Publishing Group.

https://books.google.co.uk/books?id=jKCA3leU3hMC&printsec=frontcover

Excellent answer Mr. Keith Jones...I will read that book!

Many thanks for sharing the great question Joseph! But can your question be more public to encompass almost all fields as the following: "How best can error analysis be used as a developmental tool for teachers?"?

I believe that if teacher can help students to analysis own errors it gives a good result. I am completing such a study with special reference to Geometry component.. Using meta-cognitive strategies which means that learners can identified their own errors and plan, lead, regulate and correct their own misunderstand

Dor above has hit the nail on the head as has Shiyama. I'd also add that encouraging 'error' to explore risk and experiencing it without fear is vitally important.

@shiyama: use of meta-cognition strategies to help learners identify own misconceptions and associated errors and do self correction sounds interesting.

Teacher's can use higher order questions, and scaffolds in place when the student fails to answer the questions in a manner consistent with the expected answer. This takes into consideration that learning does not occur in a vacuum and thinking itself allows for various vantage perspective answers.

I'd look at the affective domain in 'error' making. All we have to fear is fear itself :-)

These  articles could shed some light on the issue:

https://www.researchgate.net/publication/288888522_Impact_of_scaffolding_on_complexity_and_accuracy_of_Iranian_EFL_learners%27_narrative_writing

https://www.researchgate.net/publication/271881183_The_Impact_of_Teachers%27_Scaffolding_on_Iranian_High_School_Students%27_Reading_Comprehension

Article Impact of scaffolding on complexity and accuracy of Iranian ...

Article The Impact of Teachers’ Scaffolding on Iranian High School S...

Love Donna's answer and have found the following video link of the 'My favourite no' strategy to be really helpful. https://www.youtube.com/watch?v=uuDjke-p4Co 

Could also use this strategy as an exit ticket: students do a problem at the end of class and hand in their responses. Teacher picks out the favourite no to do at start of next lesson.

I teach English and I want to try this strategy with all kinds of comprehension and grammar this year.

The teacher must correct these misconceptions before entering the new lesson, and serve as a prelude to it through the entrance preparation.

Thanks Dr. Dor Abrahamson. Good Answer.

Teachers can utilize formative assessments to inform the next step in instruction. One way to encourage students who make errors in calculations is to adopt the practice of "my favorite no." The teacher looks through student errors and finds the on. By utilizing mistakes in this manner, the entire class learns from the error. They can look at their own work and examine where their thinking took them in approaching the problem. The other benefit of this practice is that teachers begin to see patterns in student responses and know how to change their instruction to anticipate common errors.