Why the answer to a question is not important to me.

In mathematics, I think the focus on getting the correct answer has permeated much of maths education, with a bit focus on tests, worksheets and textbooks in a lot of classrooms around the world it is easy to see maths as a range of questions that you need to get correct answers to.  I can tell that students believe that this is the case as often all they write in their book is the answer thinking that it is that number that I am only interested in.  

This focus on trying to find one magical number amongst all of those infinite possible answers I believe is a major contributing factor to maths anxiety.  I can see the nervousness on the faces of students as they ask me "is that right?".  I feel that if I say that it is correct then they will feel a sense of accomplishment, but the level of anxiety will come back on the very next question.  Conversely if the student gets the question wrong then their whole world tends to cave in, and you start to get comments such as "I can't do maths", "I'm stupid" and "this work is too hard" creeping in.  Their whole self-concept as a learner of mathematics tends to hinge on whether each question they come across is correct or incorrect, this self-concept can change from question to question as they get questions correct and incorrect. 

With a quick google search I find the following definitions for the word answer.  

What I find really interesting about this is that when you look at it, especially in the first definition, it mentions nothing about correctness, it simply describes an act that you do in response to a question.  It also points to the idea that an answer is more than a single statement response, it includes how you thought about it, your solution, how you thought about the problem becomes an important part of the answer.  So in this way we can define the answer to a problem as the point where you felt you could not add any more to your thinking about the problem.  The answer is not some magical number, it is where your thinking about the task stopped stopped. 

Therefore I tell my students now that I don't care what the answer is, that is not important to me, what is important is your thinking.  By looking at their thinking I can see how they thought about the problem, what mathematical thinking they may have used to solve it.  By looking at where their thinking stopped I can see where they believed was the end point of their thinking, the point where they believed that there was no more to do.  If they are multiplying fractions and have followed a correct line of thinking but have not simplified the answer it is not wrong, it just tells me that they did not recognise that they could do more thinking about the solution that they had come up with.

So why is this so important?  I have also pointed to some of the reasons such as the anxiety it causes.  But I also think it is because students believe that if they don't get the correct answer, then they haven't learned anything, the time they spent getting a wrong answer they feel is useless.  But wrong answers can teach us a lot.  Often with more complex problems mistakes are expected and celebrated as they gives us more of an insight into the problem.  If you are sitting there for an entire 90 min double lesson and working on one complex problem without giving up there is a lot of learning that has gone on there regardless of whether you have got a correct answer or not. The simple act of trying different approaches means that you are learning a lot and using a lot of mathematics in some highly flexible ways.  On multiple occasions through my teaching career I have seen a number of students get the correct answer with some incorrect thinking, and an equally high number of students get the wrong answer with highly insightful and correct thinking.

For me now it is really about changing the messages that I give to my students, it is no longer "what is your answer to the problem" it is now "can you tell me about some of the thinking you had in relation to this problem?"