What Does Show Your Work Really Mean?

Having students show their work is a struggle especially when one is told by the student "But I know how to do it" to explain the answer only.  Since I heard a college professor comment that showing work is part of finding the solution, I've come to the conclusion that having students show their work is important.

I've started telling students that by showing their work, they are actually communicating to both myself and others how they got from the question to the answer.  Showing their work is showing their journey and path.

I remember reading journals in college filled with papers written by various mathematicians.  None of those papers showed the problem with just an answer.  Every paper showed the process the mathematician used to get from the problem to the conclusion.  The work is how they communicated their thoughts, assumptions, etc to others.

Showing your work does not have to be showing every step in written form, it could just as easily include pictures showing how the students get from the problem to the solution.  I use pictures and drawings to "explain" problems to students so I don't see anything wrong with a student providing their work using drawings.

Furthermore,  I've discovered many students who showed only answers, came up with the answers in one of two ways.  First, they did the work in their heads but they were unable to explain how they got from the question to the answer.  They were unable to remember the steps they took and sometimes could not repeat the process on future problems.  They couldn't even provide any sort of illustrations to show this.

The other possibility is they did the work on an app or calculator that provided only the answer so they wrote down the answer but had absolutely no idea how the app got the answer.  This is apparent when they take a quiz or test.  Yes, I know there are apps which show the steps but most of my students tend to use Siri or other app which gives them only an answer.

Unfortunately, it is often easy to do math mentally in Elementary school but when the problems become more complex and requires multiple parts, it becomes more difficult to complete mentally.  For instance, factoring a quadratic with a leading coefficient or performing polynomial long division often requires numerous steps and is so much harder to do in your head.  The other thing, is when students continue doing the work mentally, they sometimes miss certain steps so they are unable to get the correct answer due to that gap.

I do believe it is time to get rid of the phrase "Show your work" because it is not as accurate as it might be to rephrase it into something like "Document your mathematical thinking" or "Write down notes explaining the process", or "Communicate your full solution".  Perhaps we can explain that letting us see how they got from the question to the answer is the same as writing the answer to a short answer question in English.  It is time to get past having students see "Showing your work" as a negative activity and instead see it as a positive form of communication.

I would love to hear what you think about this.  Let me know please.  Have a great day.