It is not unusual for students to experience frustration when they are asked to show their work in math. This is especially true when the problem is very simple and the answer can be memorized; how could you even explain that 5 + 4 = 9? I agree that in such cases, it is not only annoying but also unnecessary. 

However, there are many good reasons for showing work in math, and these become even more relevant in higher-level classes. As students progress through math classes into more complicated material, most questions and problems start to involve multiple steps to arrive at a solution. As a teacher, I joke with my students that I already know the answer to the question I am asking. It is not the answer that I am looking for, but to understand the process that a student followed to get there. There are often different pathways that I expect students to select, and sometimes I am surprised by an approach I had not anticipated. In class, we can discuss those different ways and analyze when using one approach is particularly helpful and when it might be too cumbersome. In this analysis, I try to emphasize self-awareness. I had a student that almost always made mistakes when using the quadratic formula, so they were advised to use that as a last resource only. 

Sometimes a student is stuck in a problem and asks for some help. As a teacher, I listen to them as they try to articulate their question. This forces them to organize their thoughts. It is fascinating to see that “a-ha” moment when they discover that they no longer have a question. It was the process of organizing their thoughts to communicate that helped them get past that hurdle. In other words, it was showing their work that answered their question.

At times a solution involves some trial and error and educated guesses along the way. I might be trying different approaches and finding none that works. After a while, I might look back at what I done before. Was there a sign mistake somewhere? If I write my work down, I can look back, verify every step and identify that small mistake. At that point, things become clear. It was not the approach that was wrong but that there was a misstep along the way. Once that is corrected, it is easy and relatively fast to finish the problem. A student that showed this work could get almost full credit in an assessment from solving a problem with only a small mistake even if their final answer is completely off.

Showing work does not mean that a student cannot skip certain steps. As a teacher and student get to know each other, a mutual understanding of what work can be reasonably skipped is usually reached. A student should also know when it is better to be more redundant and show more intermediate steps.  

Mostly, I this important for life. If my supervisor asks me to solve a problem, most likely they don’t want just a number, but an explanation of why it makes sense; what considerations were take, and why. In presenting a solution, I document and argue a logical process to arrive at a certain result. This allows for dialogue and collaboration. Those middle steps can be tweaked, and a better solution can be found and implemented. 

At the end of the day, all math is showing your work; it is organizing and documenting an unambiguous logical argument that proves a certain result. 

In short:

Math is about the process, not an end result!

Thought organization

Makes it easy to identify and correct mistakes

Communicating 

Making an organized and logical argument for or against something

Proving results