Examination of a proper solution to any problem in any domain is critical to advancment.

Examining success is a difficult skill to acquire and master. It’s not an easy task to criticize yourself when you fail, let alone when you succeed. For your time and effort, you will be rewarded with proper wisdom.

1. Finding a better solution

Everything is obvious once you know the answer and it’s path. By creating a ‘first’ path to the proof of a statement, many times, exposes new paths. Taking the time to look back at the path you took can expose logical shortcuts that weren’t obvious initially. Remapping a solution with these shorter paths will lead to shorts and undoubtably simpler solutions.

1. Solving corralaries

In the journey to solving a large and difficult proof, you are likely to come across a smaller but highly related problem to solve. For instance, in the proof of finding the length of the major diagonal of a rectangular paralelepiped given the leangth, width, and height, you’ll have to also find the diagonal of one of the faces. That corralary will lead directly to a simple solution for the larger problem. Solving this corralary can, and most likely, will help to find solutions in future problems. i.e. Finding the length of the major diagonal of a cube.

1. Building relations to existing knowledge

The most beneficial and important of the list, seek an understanding of how the arrived solution integrates with your existing knowledge. If you get into the habit of scrutinizing your own work, you will gain some form of organized knowledge.

Starting in mathematics, many students, regardless of talent, will look at some of the simplest proofs in awe. Take for instance, the well trained guitarist performing and their audience. To an unpracticed musician, the guitarist is creating, seemingly by magic, an incredible noise. To a young guitarist, there might be some insight to the foundations, but the flair is still an unknown. But to another trained ear, the steps to creating that beautiful melody is no more than combining the proper steps and motions with a bit of good timing.

In math, it’s quite similar. The proof to a statment such as, ‘The square root of 2 exists’ is daunting and unapproachable. To a student of a bit more experience, the combination of the steps will likely be more familiar. However, to a well practiced student, the steps are simply a recipe that can be repeated, simplified, and altered.

To build the familiarrity and connections between questions and their solutions is to experience. To experience mathematics is to build wisdom in your problem solving. This is your path to becoming a knowledgable mathematician and problem solver.