Best practices for showing steps in mathematical problem solving

In a previous post , I discussed the importance of showing complete steps. If you are convinced, please start to show your work while solving problems. What are the best practices to do so?

To answer this question, we must understand that the goal of showing steps is to convince anyone reading our steps (maybe your instructor, your future boss, or yourselves) that the problem is solved correctly. Thus, good work in showing steps typically has two characteristics:

• Logical Connection of Steps: All steps must be logically connected. Starting from your first one, all your steps must be logically connected. By following your steps, one should be able to see the flow of your thoughts. There are two types of steps: physical-reasoning based steps and mathematical-reasoning based steps. The first is based on physical reasoning. For example, if there is no external force, then we can conclude that the momentum of the system is conserved, and we can write down corresponding equations. The second is based purely on mathematics. For example, if we know that (A + B = C + B), we conclude that (A = C). In practice, we use both types of steps in our reasoning.
• Sufficient Number of Steps: Just enough steps are shown. There is no missing step, and there are no extra steps. Thus, jumping steps is not good – you may be able to see how you go from one to the next, but they may not be obvious to others. Showing a lot of extra information (equations that are not needed, or reasoning that is not relevant) does not help either.

Given these thoughts, the best practices are:

• Jot Down Relevant Information: Write down relevant information (equations, formulas, or reasoning relevant to the problem) on a piece of scratch paper to help your reasoning. It is OK if you miss some equations or if you have extra at this point. It really serves the purpose of facilitating your reasoning.
• Develop an Overall Strategy: From step 1, you should have a much better idea of how to execute your reasoning. Develop an overall strategy to solve your problem. Something like this: in order to get (A), I would need (B). In order to find (B), I would need (C), but (C) is already given or something already known.
• Show Your Work Step-by-Step: From the reasoning above, then we can show our work by starting from (C) to solve for (B), and using (B) to solve for (A). Use words such as “because”, “as a result”, “so”, etc., or notations such as “∵”, “∴”, “⇒”, etc., to show your reasoning. If physical reasoning-based steps are involved, explain your reasoning concisely. Note that a common error in steps is that “⇒” and “=” are used as if they are interchangeable, which is not the case. “A⟹B” suggests that if statement A is true then statement B is also true, although A and B are not equal. “A=B” means A equals B. Another common problem is that steps are all over the place, which makes it impossible to tell the logical flow of your steps. It is important to show your steps in the correct order and without any gap in reasoning.
• Verify Your Result: Now you should have a result for (A). Does the result make sense? If not, where is the error? Sometimes, it could be a simple miscalculation where we can fix the error by correcting our calculations. Sometimes, it could be something wrong in the steps. For example, it may turn out that in order to find (A), one needs to know (B’) instead of (B). In this case, we need to revisit steps 2 and 3 above.

In short, showing complete steps is very important and helpful for problem-solving, especially if the problem is relatively complex. My book contains many worked-out examples where complete steps are shown. Try to follow a similar style in showing your work. Of course, it is possible your steps may be different from mine for the same problem; however, as long as your steps are complete and make logical sense, you are doing a good job!