Looking at your own work and trying to figure out its shortcomings is a vital skill in Mathematics. This is how you iterate towards the perfect answer. The perspective that you need to analyse it from is to imagine yourself 6 months later, when you want to revise, but you've been so busy doing everything else that you've forgotten what it was that you did. You want to be able to read through your work and just be able to see exactly what it was you did, and understand why you did it. That's fairly tough to do, especially given that, while you're working out the solution, you might not understand exactly what it is that you're doing. That's why you need to make several iterations to add that understanding in. As you start to learn this skill, it will probably mean doing rough working and then writing it out in neat, but as you practice the skill more, it becomes more automatic, and you'll find yourself rewriting things less often. Warning: Often when students work in rough and write up, they leave out a lot of the detail because they are focussed on the answer. We care about the method, so it is the working that you must present, and must be aiming to refine.
This is a skill that can be quite hard to develop. It's often easier to start by looking at the work that somebody else has produced. Team up with a friend and agree that you will work on a specific question. Both of you should prepare your best solution independently. Then exchange them. Go through your friend's solution and try to understand what they did. Do you understand what they did? What specifically didn't you understand? What improvements might be made? Swap back, and try to write an improved solution based on your friend's feedback on your work, and what you learnt by reading their work.
When you go through this process, you'll be giving your friend feedback. Doing this well is a skill in itself, and there are some important ideas to keep in mind:
- Sensitivity - Try and put yourself in the position of the person receiving your feedback, and think about how it will affect them. Receiving feedback is often a daunting process because none of us like being criticised.
- Positivity - The purpose of feedback is not to rubbish somebody else's work, and to make ourselves feel better. Yes, you should point out shortcomings, but with the aim of helping to improve the work. People training you in these skills will often talk about the "feedback sandwich" - squeeze something that might be construed as negative between two positive comments. This is often difficult to do in practice, especially in maths, where about the biggest compliment you can give is simply a tick. Equally, you don't want to be so positive that the recipient thinks the criticism part is too minor an aspect.
- Specificity - It's very easy to say "I didn't understand", but that's no use in trying to improve things. Be as specific as you can about what it was that you didn't understand, and why. How might it be improved?
- Focus - If you think there's lots that can be improved, then you're probably doing a good job, but you don't necessarily have to relay all of that. If you overload the person you're giving feedback to with too many different ideas, then you risk turning the feedback into a negative process, and missing some of the important points. Identify what you view to be the most critical areas for improvement and concentrate on those.
- Privacy - While public recognition is appreciated, public scrutiny is not, so don't publicise your critique beyond the person it's intended for, whether this is in written form, or verbally.
- Opinion - What you are conveying is often your opinion, not an absolute fact. If so, make sure you phrase it as that: say "I didn't understand xxx" rather than "xxx was complete nonsense". Try not to exaggerate to make a point. Avoid words like "never", "all," and "always" because the person will get defensive.